help-49

[cinder flax]

so cos2x = ?

[small crypt]

no

cosx

[cinder flax]

i solved it alr last year

[small crypt]

gr8

[cinder flax]

there is a formula

cos2x = cos^x - sin^x

[small crypt]

yeah

so you write it as

[ (1- cos2x)/2 ]^5

the same thing

but

in the middle

[cinder flax]

u also have cos2x = 1 - 2sin^2x and 2cos^2x -1

[small crypt]

ryt

exactly youd have to use tht

by solving after applying binomial exp

you'd get cosx = + - root 3 by 2

[cinder flax]

umm there is a simpler way

[small crypt]

wht

[cinder flax]

find sin^2(X)

and cos^2 (X)

[small crypt]

oh

but i found that simpler

as binomial is easy for me

like the application

since all odd power terms cancel

[cinder flax]

hmm alr lets go with that

[small crypt]

so ans is n(pi) + - pi/3

[cinder flax]

uhhh no

[small crypt]

wait wht

[keen mauve]

Graphically sin^10x+cos ^ 10 x is symmetric about 1/2 min value is 1/16 and max value is 1

[small crypt]

lol

that aint required

[cinder flax]

what is ur eq, b4 the binomial

[keen mauve]

So solve 29/16 cos 2x power 4 range is 1/16 to 1

[small crypt]

lemme write and show

[cinder flax]

ok

[small crypt]

here

@cinder flax

do the expansion yourself

[cinder flax]

yes exactly

[small crypt]

so

ans is n pi +- pi/3

[cinder flax]

i thought u were doing binomial se

[small crypt]

accha

where do you live btw

delhi?

or kota

xD

[cinder flax]

delhi

what kind of que is that tho TT

[small crypt]

trig equations

[azure bolt]

whats the question

[small crypt]

written

on paper

[azure bolt]

where

[small crypt]

at the top

scroll up

[midnight plank]

@small crypt Has your question been resolved?

[cinder flax]

i think its solved right?

[cinder flax]

this is correct

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[modest wasp]

How to factorise 1 minus 2(sin .cos)/1+2sin.cos to 2sec.csc

[last slate]

nooooo

[modest wasp]

Like I was able to do rest of trig but I always get stuck at last step of factorization

[rain wasp]

$\frac{1 - 2\sin x\cos x}{1 + 2\sin x\cos x}$

[rain wasp]

are you asking how is this reduced to $2\sec x\csc x$?

[modest wasp]

Yes that's the one

[modest wasp]

Yes

[last slate]

magic

wizard

how

[dreamy lichen]

it doesnt seem like the same thing

plug in 0

its not even same at 0

[modest wasp]

Hold on then let me recheck i think this was the q ... Let me recheck

[rain wasp]

if possible, please provide a screenshot

[dreamy lichen]

now this is different lol

[modest wasp]

I got till 1 minus 2sin.cos whole divided by 1+2sin.cos

[rain wasp]

[dreamy lichen]

How?

[rain wasp]

can you show your work?

[modest wasp]

Lcm then identity of a plus b whole square in numerator and in denominator a minus b whole square

Holy wait I'm dumb wrong lcm

Wait

[dreamy lichen]

it still doesnt look the same

[dreamy lichen]

this is still false, evaluate it at pi/2 lol

what kind of question is that

[modest wasp]

The q ain't right?

[dreamy lichen]

seems like it

[modest wasp]

Bruh chatgpt making wrong qs I just wanted to practice my weakest ch

[dreamy lichen]

are you sure you are proving the identity? Arent you supposed to solve it as an equation?

[dreamy lichen]

okay, that explains it

just find some questions on the net

"trig identities worksheet"

search that

if you want, you can add "filetype:pdf" to only search pdfs

[modest wasp]

No it was to be proved guess screenshot trimmed that part

[dreamy lichen]

chatgpt is still shit at math

just use human-made worksheets

they can also be wrong, but not as often as chatgpt (which takes the question from those worksheet anyway...)

[modest wasp]

Idk man I'm at 10th grade math whom help should I take

[dreamy lichen]

^ search this

u should find plenty of worksheets

[modest wasp]

Thanks

Imma try these q

[dreamy lichen]

and the pdf i took it from even has solutions

https://teaching.martahidegkuti.com/shared/lnotes/5Trig/identities1.pdf

Open at your own risks, pdf's can be dangerous (though i dont think its the case here)

[modest wasp]

Thanks again

How to simplify (a + b)/b

[dreamy lichen]

a/b + b/b

[plain karma]

a/b + 1?

[dreamy lichen]

in general
(a + b) / c = a/c + b/c

[plain karma]

(a + b)/b is already okay

[dreamy lichen]

note that if the sum is in the denominator, you cant do the same
c / (a + b) IS NOT c/a + c/b

[modest wasp]

Guess it didn't work I'll have to simplify more man i suck at factorization how to do factorization

[dreamy lichen]

what are you simplifying now

what identity are u proving

[modest wasp]

Sin(1+sin)+cos² whole divided by cos(1+sin) to 2tanxsecx

Wait why I got sec i must've did something wrong

What the nvm the q had subtraction I was doing addition

It was practice problem 3 I'll try again

[dreamy lichen]

this one i suppose?

you probably went wrong somewhere

[dreamy lichen]

this is already wrong

[modest wasp]

Yes

I noted the q wrongly

[dreamy lichen]

btw if you are reasonably good at typing math on computer, you can try solving it in desmos or sth like that

if you plot both the RHS and LHS, the graphs should overlap

and whenever you make a mistake when simplifying it, you'll see it because the graphs will no longer overlap (such as green with red here)

that way you'll see immidiately where and when you make a mistake

[modest wasp]

Thanks I'll do this to know what I am doing wrong

Also any factorization tips

I'm bad at it

[dreamy lichen]

well the most important factorizations are
a^2 - b^2 = (a-b)(a+b)
a^2 + 2ab + b^2 = (a+b)^2
a^2 - 2ab + b^2 = (a-b)^2

the first one will probably be most common

and then obviously if its something like
2sin(x)cos(x) + sin(x), you can factor sin(x) from both of them and get
sin(x)(2cos(x)+1)

so factoring out the highest common factor

[modest wasp]

So just identities and seeing what is common

[dreamy lichen]

yep

[modest wasp]

Thanks .close

How to close

[dreamy lichen]

literally .close

just type it in individual message

[modest wasp]

.close

[midnight plank]

Channel closed

Closed by @modest wasp

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[gaunt jetty]

<tikz>
\lm Hey, I think I am messing up simple geometry with this one, but is it not the case that the minimum chord distance between the inner points is $\32\,A_2$ in the following?
\begin{center}
\begin{tikzpicture}
    \def\R{3}
    \def\r{1.2}
    
    \draw (-3.5,0) -- (3.5,0);
    \draw (0,-3.5) -- (0,3.5);
    
    \draw (0,0) circle (\R);
    \draw (0,0) circle (\r);
    
    \foreach \angle in {0, 90, 180, 270} {
        \fill (\angle:\R) circle (2.5pt);
    }
    
    \foreach \angle in {45, 135, 225, 315} {
        \fill (\angle:\r) circle (2.5pt);
    }
    
    \draw[-latex] (0,0) -- (20:\R) node[midway, below right] {$A_2$};

    \draw[-latex] (0,0) -- (135:\r);
    \draw[dotted] (135:0.8) -- ++(210:1) node[dotted, anchor=south east, inner sep=1pt] {$A_2 \4{\32}5$};
\end{tikzpicture}
\end{center}
I used the formula $d_{\t{min.}} = 2R\6\sin{\5\pi M}$ where $M$ is the amount of points

[midnight plank]

@gaunt jetty Has your question been resolved?

[gaunt jetty]

.close

[midnight plank]

Channel closed

Closed by @gaunt jetty

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[wooden fossil]

combinatorics exercise I am confused with:

There are n people sitting around a round table. x ordered meal 1, y ordered meal 2 and the remaining people meal z.
How many ways are there to arrange the meals around the table?

I would've said that since we can most likely distinguish the people there are (n choose x) + (n choose y) ways.

But i am really confused with this exercise. Does anyone know more?

[lyric charm]

x ordered meal 1
what does that mean

this seems to be a language issue

[wooden fossil]

So the n people are at a restaurant and each person orders a meal.
x of the n people decide to order the first meal @lyric charm

[lyric charm]

oh ok

why are the meals called "1, 2, z" rather than 1, 2, 3

so we have x orders for meal #1, y orders for meal #2, and (n-x-y) orders for meal #3? is that correct?

C(n,x) + C(n,y) is wrong; this means choosing ONE of the first two dishes and then choosing who at the table gets it.

[midnight plank]

@wooden fossil Has your question been resolved?

[wooden fossil]

uhm my brain malfunctioned

Im sorry , it was supposed to be 3

[lyric charm]

consider thinking of the setup this way: you first bring the x orders of meal #1 and have C(n,x) ways to distribute them

then you bring the y orders of meal #2 and there's now (n-x) people you could serve them to...

[wooden fossil]

so you say it should be C(n,x) + C(n-x, y)? @lyric charm

actually thats kind of what I think I was trying to say

but again my brain didnt brain Im sorry

I was especially confused because the students that get these exercise havent been introduced to the binomial coefficient yet. So I thought this should be countable by more basic techniques. But ig the binomial coefficient can also be discovered here as it is just a way to use the more basic principles

in any case thanks a lot!

[midnight plank]

@wooden fossil Has your question been resolved?

[lyric charm]

no, times, not plus

[lyric charm]

... what grade level

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[hoary hamlet]

can someone explain this question intuitively?

[lyric charm]

you have 10 red numbers and 20 blue numbers. the red mean is x and the blue mean is x+4.

[hoary hamlet]

yep

[lyric charm]

one of the red numbers and one of the blue numbers swap places

the new red mean is y and the new blue mean is y+1

you're asked for y-x

[hoary hamlet]

okay im following

[lyric charm]

well, there's your explanation of the question

do you think you have an idea now for what to grab onto

[hoary hamlet]

i unfortunately dont, i was thinking that whatever 2 numbers were swapped caused blue's mean to drop by 3 but that's clearly not the case

[lyric charm]

ok then lets try to think a bit about means generally

if you know a data set's mean and sample size, then what else can you get easily

[hoary hamlet]

the total?

[lyric charm]

indeed.

you have 10 red numbers and 20 blue numbers. the red mean is x and the blue mean is x+4.
can you give an expression in terms of x for the total of all the red and blue numbers involved

[hoary hamlet]

that would be 30x + 80

oh and then the total would be conserved across the swap

[lyric charm]

that it would

[hoary hamlet]

so we can just form an equation and find y-x

genius

lemme try

yep got it

thank youu

.close

[midnight plank]

Channel closed

Closed by @hoary hamlet

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[wooden fossil]

sorry, was afk. Its undergrad university

[wooden fossil]

but anyways

yes mutliplication makes much more sense... Idk why Im acting like an idiot rn sorry

and again thanks a lot. as always @lyric charm highly appreciated!

.close

[midnight plank]

Channel closed

Closed by @wooden fossil

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[twilit field]

Just wondering if equipartitions are the most ideal here

ooh,

got it

.close

[midnight plank]

Channel closed

Closed by @twilit field

Use .reopen if this was a mistake.

[cerulean oyster]

half area of a parabola, right?

[twilit field]

yea, but I can't use FTC

I have to use the definition

[cerulean oyster]

im to engineering pilled i would have just appealed to the geometric formula, lmao.

[twilit field]

😭

.reopen

[midnight plank]

✅ Original question: #help-49 message

[twilit field]

What I basically have to do is use this:

[cerulean oyster]

yeah, ik, riemann sum

[twilit field]

.close

[midnight plank]

Channel closed

Closed by @twilit field

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rapid rivet]

Hi I need help with a question (my own translation from my native language)

The amount of a radioactive substance decreases according to the function N(t) = N(arrow down)0 e^kt where N is the amount of the atom nucleuses efter t days. The measurement starts when the substance has 4,0 • 10^21 atoms. Then the change of atoms is -0,4 • 10^21 per day. Decide the function N(t)

[worthy tundra]

you got the original one?

[rapid rivet]

Yeah

[worthy tundra]

maybe send the original one thanks

[rapid rivet]

But you wouldn’t understand Swedish?

[worthy tundra]

is ok just send first

[rapid rivet]

@worthy tundra did it help?

[worthy tundra]

yeah actually

1. next time dont worry, there's google lens to help us translate. the concern is sometimes when you translate, i'm not too sure of the math symbols (e.g. what your arrow down meant)

2. okay here's my thought process for the solution, let me guide you through it

so based on the question, we know that:

• the formula has a N。-> this refers to the number of atoms / whatever at the start (when t=0, hence that subscript 0). this number is 4.0 x 10^21
• the rate of decrement is -0.4 x 10^21 per day

any confusion so far?

[rapid rivet]

Nope, that I understand

[worthy tundra]

3. okay, so we know when t=0, N(t) should give us 4.0 x 10^21 too, since time hasn't passed, there should not be any decrement.

N(0) = (4.0 x 10^21) e^(k * 0)

k * 0 = 0, e^0 =1, so far making sense too

4. so now calculate for yourself, what happens when t = 1?

• what is your N(1) based on the rate of decrement?
• can you derive e^kt?
• and from the above, can you then determine what k is?

@rapid rivet any confusion?

[rapid rivet]

My bad I’m just very slow

[worthy tundra]

speed doesnt matter, im more concerned if there are things you cannot get why

do you understand part 3?

[rapid rivet]

I’ll just use ln to get K no?

[worthy tundra]

yessir!

[rapid rivet]

Since e gets removed

Bomba I understand it now

Alr let me try tho

[worthy tundra]

to make sure it's the right answer, you could always calculate N(t=2) to check

[rapid rivet]

Do I need to get ln on the 4.0•10^21 or is the e^k enough?

[worthy tundra]

hold up

what are your steps so far

so what is N(1)?

[rapid rivet]

N(1) = (4,0•10^21) • e^k

[worthy tundra]

i mean the ACTUAL value of N(1)

you can find that out based on what the question already gave you :)

[rapid rivet]

4,0•10^21 - (0.4•10^21) ?

[worthy tundra]

that is?

[rapid rivet]

Lemme get a calc

[worthy tundra]

HEY HEY

you dont need a calc for that

4,0 - 0,4 = ?

[rapid rivet]

3,6

[worthy tundra]

so back to this

what is e^k?

[rapid rivet]

-0,4•10^21

[worthy tundra]

u sure....? 🤨

[rapid rivet]

No

Yes it is no?

🙂

[worthy tundra]

3,6 = 4,0 * e^k

[rapid rivet]

So it’s 0,4

[worthy tundra]

how did you get 0,4

4,0 * 0,4 = 1,6 not 3,6

[rapid rivet]

I’ll just do 3,6/4

[worthy tundra]

so that gives you?

[rapid rivet]

0,9

[worthy tundra]

so e^k is 0,9 am i right?

so what do you ln on?

[rapid rivet]

e

And other side

[worthy tundra]

what other side

[rapid rivet]

ln 0,9 = k?

[worthy tundra]

yess

[rapid rivet]

ln 0,9=-0,105

[worthy tundra]

so depending on your teacher, you can either leave it as ln(0,9) or you can leave it as 3 decimal places form (-0,105)

so re-expressing N(t) with your calculated k

you will have N(t) = N。e^(-0,105 t)

[rapid rivet]

Yes

[worthy tundra]

i think you should be fine generally, just dont jump to conclusions yeah? be careful

:)

[rapid rivet]

That’s the same as N(t) = 4,0•10^21 • e^(-0,105)•t right?

[worthy tundra]

yes, but we generally don't want to replace N。

[rapid rivet]

Ohhh alr

[worthy tundra]

because that can be an arbitrary number

the starting number can change too :)

[rapid rivet]

True that

[worthy tundra]

that's why we dont fix it to 4,0

[rapid rivet]

Well many thanks for helping me

[worthy tundra]

np!

[rapid rivet]

I see you have the chess role, maybe we could play sometime (I’m better at the chessboard than next to a mathbook) 🙂

[worthy tundra]

hahaha i don't really play chess anym, just occasionally looking at some puzzles

have fun math-ing though!

you can do this

rmb to close the chat once you're done @rapid rivet

[rapid rivet]

Yo can I just ask you one more math question

There’s 1 more question that I didn’t get

[worthy tundra]

sure i'll try

[rapid rivet]

c

[worthy tundra]

give me a while to write the stuff yeah

you manage to get the solution but don't know why or?

you can't solve it at all?

[rapid rivet]

I mean idk if I can do it the same way as with b

[worthy tundra]

yeah you can, it's the same rule as the one in b

but you need to manipulate it slightly

[rapid rivet]

In what way

[worthy tundra]

how would you solve it? maybe i'll let you know what went wrong based on your working

[rapid rivet]

How to do root on keyboard tho?

[worthy tundra]

sqrt()

[rapid rivet]

Ohhh ok

[worthy tundra]

or if you want you can use latex $\sqrt{x}$ $\frac{1}{\sqrt{x}}$

[grand pond]

min2

[rapid rivet]

(1/ k•sqrt(x+h) - 1/k•sqrt(x)) • (1/ k•sqrt(x+h) + 1/k•sqrt(x)) / h (1/ k•sqrt(x+h) + 1/k•sqrt(x))

That’s how I’d start

[worthy tundra]

wait WHERE DID H COME FROM

[rapid rivet]

Wait my bad

That was supposed to be in the bottom part only

No wait

No it’s supposed to be like that no?

[worthy tundra]

you're working on question 2247(c) right?

[rapid rivet]

Yes

Like this

[worthy tundra]

1. okay we take a look at (b) first

$f(x) = \sqrt{x} = x^{\frac{1}{2}}$

$f'(x) = \frac{1}{2} * x^{-\frac{1}{2}}$

[grand pond]

min2

[worthy tundra]

this is based on the rule where d/dx of x^n is n * x^(n-1)

2. now the similar rule can be used for (c) but we need to put it in the "same format" first

[rapid rivet]

Wait can I show on paper how I’d solve B?

[worthy tundra]

uh shure

[mystic fulcrum]

I think you need to go through the definition of derivatives

[worthy tundra]

you mind helping to paste that huge chunk of rules here?

iirc theres some shortcut or something

[mystic fulcrum]

The lim h->0 thingy

[worthy tundra]

HUH

[mystic fulcrum]

Uhhh idk how to use latex im rly new here

[worthy tundra]

hes working on 2247 not the limits one

[mystic fulcrum]

Yeah is saw that in the title it says derivative definition but i may be wrong cus i dont know that language

Wait i mean like this let me send a pci

[worthy tundra]

$f(x) = \frac{1}{k \sqrt{x}} = \frac{1}{k} * \frac{1}{\sqrt{x}}$

[grand pond]

min2

[worthy tundra]

oh yea

i gets what you mean by

[mystic fulcrum]

Like finding it through the limits instead of using n*x^(n-1) rule straightly

[worthy tundra]

the definition part

missed that earlier

[rapid rivet]

That how I’d solve b

But idk how to remove the K on c

Brb I’ll just help my father with something

Back

[rapid rivet]

Wait how to get rid of the K in this case (c)

[fathom onyx]

And... you want to get rid of the k, why?

[mystic fulcrum]

I have no idea

[rapid rivet]

I mean I want to do kinda like this

[fathom onyx]

What's stopping you from following the same procedure?

[rapid rivet]

With 1/k sqrt(x)

[rapid rivet]

I will get k as well no?

[mystic fulcrum]

I think the numerator is multiplied by 1/k maybe u can try to bring it out

[fathom onyx]

yes to both, and I don't see why that should still be a problem

(remember, you're introducing a dummy variable h in the limit)

[rapid rivet]

I’ve just never done it with so many variables so idk if that’s how it’s supposed to be

[worthy tundra]

i- can't help you further im sorry my D/I knowledge is VERY rusty

[midnight plank]

@rapid rivet Has your question been resolved?

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rapid rivet]

Thanks for helping tho

[fallow scarab]

Can you repost your question since your previous one timed out

[midnight plank]

@rapid rivet Has your question been resolved?

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[coral belfry]

,tex
Hello. So the following stuff is a proof our professor wrote in class. I wrote what we are proving at the start. The proof isnt complete, because i have a problem with what he wrote in the end of the things i wrote. Did he get confused, or am i not seeing why what he said is true?
\
\
"For all $x_n $ sequences, with $ x_n \in D_f , x_n \neq x_0 $ and $\lim{x_n} = x_0 $ , then $\lim{f(x_n)} = l $" $\implies $ "$ \lim{f(x)} = l , x\to\infty $"
\
\
Proof: We will use the fact that $\neg \left( P\implies Q \right) = P \ \text{and} \ \neg Q $
\
\
So, assuming the negation, we have:
\
$\forall x_n , x_n\in D_f , x_n \neq x_0 , \lim{x_n} = x_0 $ :
\
\
$\lim{f(x_n)} = l , \ \textbf{and} \lim{f(x)} \neq l $
\
$\lim{f(x)} \neq l \implies \exists \epsilon > 0 $ s.t. $ \forall \delta >0 , \exists x $ s.t. $ \lvert x-x_0 \rvert < \delta $ and $ \lvert f(x) - l \rvert \geq \epsilon $
\
\
Let $ \delta = \frac{1}{n} , n\in\mathbb{N} $. So: $ \exists x_n \in D_f , x_n \neq x_0 $ where
\
$ \vert x_n -x_0 \rvert <\frac{1}{n} $ and $\lvert f(x_n)-l \rvert \geq \epsilon $

[grand pond]

fijokazż

[spiral rock]

what is the issue

aside the symbol overload

[coral belfry]

at the end, why do we have that |f(xn)-l | >= ε?

[coral belfry]

thats my doing

[spiral rock]

thats from the negation of lim f(x) = l

[coral belfry]

but then we shouldnt have x_n inside f

just x

[spiral rock]

thats how he calls the x that you know exists for delta = 1/n

[coral belfry]

how does he just call it x_n ?

ik x_n is entirely in Df but that doesnt mean that whats true for some x must be true for some x_n too

[spiral rock]

x_n is just a name

[coral belfry]

x_n is a sequence

[spiral rock]

right, its just this name is not used at this point of the proof

he could have said, choose x for n=1, choose y for n =2 etc

but then he needs infinitly many symbols

so he gets infinitly many symbols by using subscripts

also this hints on what he is going to do with the x_n

[coral belfry]

hmm maybe i understand

so like

we have |f(x) -l | >= epsilon

for x within delta of x_0

and we just write x as x_n

i mean that sounds sloppy

ohh

is it because x_n converges to x_0?

[spiral rock]

thats the next step of the proof

[coral belfry]

huhhh

but if we dont use the fact that x_n converges, so x being within delta of x_0 means x can be written as x_n

its like we just say stuff without proof

[spiral rock]

x_n is a name

[coral belfry]

that seems rlly absurd to me lol

we defined x_n to be any sequence that converges to x_0

so it cant be also just a name

[spiral rock]

note that x_n is not already used at that point

its only used in quantifiers

[coral belfry]

but we have it as initial condition

our P is the second thingy not the first

sorry shit

i meant our P is the first sentence at the top

[spiral rock]

does it make more sense if you call it y_n instead?

[coral belfry]

yeah, as long as its not connected to x_n

[spiral rock]

sure so you can do that

its just a rename

[coral belfry]

okay so

we wrote "there exists" a y_n with that property

nvm i dont got a problem with that

so we use sandwich

and y_n converges to x_0

then we use the fact that lim(x_n) = x_0 => lim(f(x_n)) = l

but can we really use y_n instead of x_n here? is it because we showed that it converges to x_0 ?

if thats right then i think i get it

[spiral rock]

yes, because x_n is any sequence that converges to 0, and y_n is a specific sequence that converges to x_0

[coral belfry]

hmmmm okayokay thank you for recommending a different variable for it

.solved

[midnight plank]

Channel closed

Closed by @coral belfry

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[leaden seal]

is the bit in red correct?

cancelling the terms

[midnight plank]

@leaden seal Has your question been resolved?

[olive matrix]

yes that seems fine

[leaden seal]

woudnt it be Ps/Eb = Pb/5As

[olive matrix]

why?

[leaden seal]

are we not diving them?

[olive matrix]

dividing what

to go from the top line to the bottom line, we are multiplying both sides of the equation by Eb As

[leaden seal]

ah yh okay

[sonic sun]

Looks correct

[leaden seal]

thanks

.close

[midnight plank]

Channel closed

Closed by @leaden seal

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[midnight plank]

Channel closed

Channel closed due to the original message being deleted.
If you did not intend to do this, please open a new help channel,
as this action is irreversible, and this channel may abruptly lock.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[robust field]

I’m not sure

[quick creek]

0.02×F=0.15×20+0.33×120

Σt=0

use the torque

[midnight plank]

@robust field Has your question been resolved?

[robust field]

I don’t understand

<@&286206848099549185>

[surreal charm]

Hi

[robust field]

Hi

[surreal charm]

Use this

[robust field]

I’m not able to start

[surreal charm]

To balance all forces

[robust field]

Wait

[surreal charm]

Ok

Do you know how to calculate moment of a force

[robust field]

F x d?

[surreal charm]

Its perpendicular distance × F

[robust field]

Oh I get it

I didn’t read the question

P is the pivot

[surreal charm]

Right

[robust field]

f should be equal to the both downward force

Damn

If you know what I mean

[surreal charm]

Yes torque by f should cancel those

[robust field]

What is torque

[surreal charm]

Basically you can say moment

[robust field]

Ok

How to do

ii)

[surreal charm]

I'm not sure what do they mean

You have F

[robust field]

Exactly

[lyric charm]

did you do (i)

@robust field actually do you still need help with this

[midnight plank]

@robust field Has your question been resolved?

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[hoary hamlet]

how does one answer this question

[mortal falcon]

first, write out the inequalities and information given

then do algebraic manipulation to arrive at one of the given ineualities

you're given 2 statements, so start by translating them into symbols

[hoary hamlet]

heres what i got so far.

i have no clue how to o

go on from here

[remote mauve]

can you write that out pls in text or latex

I also have bad handwriting dw

oh nvm

[mortal falcon]

one thing you can do to rule out possibilities is look for counterexamples

[remote mauve]

R+H<10, 2pir(r+h)<50

got it

[mortal falcon]

for instance, can you find values for r and h such that your two inequalities are true, but r + h >= 8?

[remote mauve]

yes you can prolly do that proof by contradiction ig

[hoary hamlet]

my handwriting isnt that bad i swear 😭

[remote mauve]

But I think you should prove it

[remote mauve]

I read it dw

[hoary hamlet]

this question is supposed to take

like

under 4 mins

[remote mauve]

mines far worse

[mortal falcon]

there is no difference

[remote mauve]

during an school test there is

[hoary hamlet]

would i have to trial and error the 2 equations

[remote mauve]

nah

[mortal falcon]

"prove it" doesn't mean anything for this to be in disagreement with "find a counterexample"

and in general finding counterexamples can be a simple sanity check on a test

[hoary hamlet]

i have one for r=1, h=8

so that rules out II

interesting

oh that also rules out II

III

using the same variables

[remote mauve]

5r+5h<50, 2pir^2+2pir<50

[mortal falcon]

yup, so that only leaves I

[hoary hamlet]

but eventually id have to prove one of them right

unless none of them

are true

but the answer is that I is true

[mortal falcon]

yeah, you still have to verify if I is true or false, and that will require algebra if it is true

[hoary hamlet]

ive been trying for a while

cant come up with anything

😭

[mortal falcon]

since $r(r+h) < \frac{25}{\pi}$ from inequality 2, $h < \frac{25}{\pi r} - r$, so $\pi r^2 h < \pi r^2 (\frac{25}{\pi r} - r) = 25r - \pi r^3$. \

You can maximize this expression with respect to r through a simpler derivative computation, and the resulting max volume is $\frac{250}{3 \sqrt{3 \pi}} = 27.1 < 65$, so I is true

oh i should ask first if calculus is allowed

[hoary hamlet]

yeah

it is

[mortal falcon]

okay then yeah

you just take a derivative, find roots, maximize, and then check that the upper bound is less than 65

[hoary hamlet]

lemme try

[grand pond]

snowflake

[hoary hamlet]

how does differentiating an inequality work?

[mortal falcon]

you're not differentiating an inequality

you have an upper bound for pi r^2 h, and you want to show that the maximum value of that upper bound is less than 65

[mortal falcon]

sure, you want to find the maximum value of that, but you want it to be in terms of r only

so you have to find an upper bound for that that replaces h in terms of r

[hoary hamlet]

yeah yeah

from line 3 i have the equation

and i want to maximize pi r^2 h

right

[mortal falcon]

you just want to find a concrete upper bound for pi r^2 h

and a way of doing so is maximizing the upper bound in line 3

[hoary hamlet]

yeah but it would be easy for me if it was an equation but im just confused on how its going to work if its an inequality

does the inequality stay?

[mortal falcon]

i dont think you understand why we're doing this

[hoary hamlet]

like this?

[mortal falcon]

we have a quantity, let's call it $V(r, h) = \pi r^2 h$

[grand pond]

snowflake

[mortal falcon]

we just want a constant upper bound, some number K such that $V(r, h) \leq K$ for all valid $r, h$

[grand pond]

snowflake

[hoary hamlet]

yeah i get that

its like finding the maximum point of a quadratic

and that maximum point becomes the upper bound

so that if that upper bound is less than 65

then the inequality holds true for all r, h

yeah?

[mortal falcon]

this is kind of unclear out of context but in a vague sense sure

I think what you're describing is say, maximizing $V(r, h)$ directly, and letting that maximum be the upper bound

[grand pond]

snowflake

[hoary hamlet]

yeah

[mortal falcon]

the issue is just that doing that maximization is hard, and we don't actually care that much about finding the maximum of $V(r, h)$

[grand pond]

snowflake

[mortal falcon]

so instead, we can search for a function $f(r)$ such that $V(r, h) \leq f(r)$ for all $r, h$. \

Then, instead of maximizing $V(r, h)$, we can just maximize $f(r)$. This only requires optimizing 1 variable instead of 2, which is much easier. \

If we can find $C$ such that $f(r) \leq C$ for all valid $r$, then $V(r, h) \leq f(r) \leq C$ for all valid $r, h$, making $C$ a valid upper bound

[hoary hamlet]

ohhhh

so we maximize this

[mortal falcon]

exactly

[hoary hamlet]

thats what i was confused about with the inequalities.

so if the inequality was the other way around

this would not be valid?

[grand pond]

snowflake

[mortal falcon]

like, maximizing a lower bound would not give us an upper bound, yeah

we want to find the worst case for our upper bound basically, to guarantee that it's valid

[hoary hamlet]

so the value you obtained in your derivative computation, the ACTUAL value of V(r, h) will never actually reach that value?

[mortal falcon]

yes exactly

i called it max volume that's my bad

[hoary hamlet]

i see now

lemme try

[mortal falcon]

it's just a max on the volume

I do a see a much simpler argument now that doesn't need calculus 😭

[hoary hamlet]

its fine i think this makes a lot of sence

sense

alrighty thanks for your help

.close

[midnight plank]

Channel closed

Closed by @hoary hamlet

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[eternal moss]

Can anyone help me with this ?

[vivid yoke]

,rccw

[mortal falcon]

what progress have you made so far? what angles have you found

[eternal moss]

No u just need to find the angle x

Is it 99 or 25 ?

[mortal falcon]

okay, so you aren't going to be able to find x without finding other angles first

[ocean wren]

DDo you notice how yo7 can f8nd angle ABH?

[surreal charm]

can you type slow

[ocean wren]

I should indeed type slowly 😭

[mortal falcon]

reiterating, what progress have you made so far? what angles have you found in the diagram so far

[ocean wren]

alright so uh here look. The blue part is a construction to make the explanation better but it isn't needed

since ABC is a straight line

don't angle 1 and angle CBH make a linear pair?

Where did bri go

Bro*

😭

[midnight plank]

@eternal moss Has your question been resolved?

[eternal moss]

Bro srry

The thing is

This is my exam question

The answer i wrote was 99

Which our teacher wrote as wrong

And my other class mates put 25

Which was correct

But I dknt understand how is it 25

We know that F is corresponding to B which is 53 so F is 53

And since it's on a straight line its 180

So 53 + 29 + x is 180

[eternal moss]

28*

Which is 81

So 81 minus 180 is 99

But our teacher told the answer is 25

So can anyone help me to tell how is 25 the answer

[lethal path]

no, angles on a straight line refers to only 2 angles

like this

[eternal moss]

Isint a straight line 1?80

Becuase that is what is i learned

[warped nest]

Yes, but you need a line to go through that line to create that rule

Angle on a straight line

[warped nest]

Here you need to notice the fact that there are 2 triangles, i think

[lethal path]

I don't think you know what the correct statement was

anyways, what is angle EFH from this info?

[eternal moss]

Uh

[ocean wren]

YYou don't have to use two triangles.

[warped nest]

You know that:
180 - 53 = 127 (Angle at ABH)
Lines ABC and DEFG are parallel, so you can use alternate interior angles, meaning the corresponding angle at F is also 127

[eternal moss]

The angle between F and E is 127 my teacher said

[warped nest]

maybe, there are many ways to do these types of questions

[eternal moss]

Wait so u got 127 now so now what

[ocean wren]

way too many ways

[ocean wren]

now you use the angle sum property of a triangle.

[eternal moss]

...

Hoa do u do thay

[ocean wren]

no no wait

[warped nest]

yes

[ocean wren]

you need angle BAE

[warped nest]

in a triangle, the 3 angles must sum up to 180

[ocean wren]

di you know how to find angle BAE?

[eternal moss]

127 + 28 is 155

[ocean wren]

ye

[eternal moss]

155 -180 is 25

Oh I get it now

[warped nest]

other way round

[eternal moss]

So im wrong

[warped nest]

but yes

[eternal moss]

Oof

[warped nest]

yes you were wrong

well its not a big deal

[eternal moss]

Welp I couldn't get my marks

[warped nest]

i do much harder maths and these i cannot do for the life of me

[eternal moss]

If I would have gotten that

I could have gotten 90 %

But welp

[warped nest]

meh, dont feel so sorry about that

[lethal path]

it's okay you'll be doing plenty more tests in future

[eternal moss]

Ty for the help

[warped nest]

what i suggest though

for these types of questions

[lethal path]

you can bounce back

[eternal moss]

Ye

[warped nest]

is understanding how the angles work in terms of naming

For example

you don't say "angle B", you say the angle at ABH

[eternal moss]

Ohhh

[ocean wren]

angle ABC ≠ angle ACB

[eternal moss]

Ok

[lethal path]

yes, there's two different angles with vertex B

[warped nest]

the middle letter is the important one exactly

[lethal path]

so you have to go, C to B to H

angle CBH

then that tells you exactly which angle

[ocean wren]

but. Amgle ACB = angle BCA

[warped nest]

yep exactly, if you focus on this you'll find these much easier

[eternal moss]

Well ty guys

[ocean wren]

I meant

[lethal path]

if you literally connect those points with your pen that's the angle

[ocean wren]

close?

[eternal moss]

How

[ocean wren]

,close

do tha

[eternal moss]

,close

[warped nest]

with the .

[ocean wren]

oh it's a period

[eternal moss]

.close

[midnight plank]

Channel closed

Closed by @eternal moss

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[upbeat yew]

what does the heine borel theorem mean?
when to use it when solving mathematical calculus equations?

[midnight plank]

Please don't occupy multiple help channels.

[upbeat yew]

When I'm deriving an equation in advanced calculus, there is an option of using the heine borel theorem for shortcutting the steps or proving something but I'm unsure on how to do it

and there is that lhopital rule and chain rule for derivation which is something also confusing to me

why would you do lhopital theorem for limit finding of a function/continuous series?

[prime hornet]

the Heine-Borel theorem is a topological statement

all it says is that any subset of R^n is compact if and only if it is closed and bounded

I’m not exactly sure how this is used to do anything with calculus

can you show an example of what you mean?

[upbeat yew]

yes

finding max and min for function:
f(x) = x*sqrt(4-x^2)
in the set of [-1,2]
using heine borel theorem

[upbeat yew]

and there are more examples of using it on derivatives etc.

[prime hornet]

HB isn’t the main thing you’re using here

all it tells you here is that [-1, 2] is compact, and so you know there must be a max/min for f on that interval because of the extreme value theorem

actually computing the max/min is a very different story from just knowing they exist

[upbeat yew]

we see that interval is [-1,2] is compact and closed (by heine borel), and thus weirstrass theorem says it must have maximum and minimum

[prime hornet]

sure, but that won’t tell you what the max/min actually is

[upbeat yew]

finding the max/min is using the derivative

[prime hornet]

correct

[upbeat yew]

so we use heine borel to say that interval is closed and bounded (compact), thus it must have a min/max.
Afterwards, we use derivation-- (i.e. chain rule) and compute using a table or something the min/max by: = 0.

[upbeat yew]

but my question is, finding the min/max and using the heine borel theorem and then using lhopital rule is very confusing to me
When do I use the lhopital rule?

[prime hornet]

you already seem to know how to find maxes/mins, so I’m not sure what’s confusing about that

[upbeat yew]

the lhopital rule

and I'm afraid of min/max miscomputations

[prime hornet]

the HB theorem is not used to find them, it’s just something you use to be able to say a set is compact or closed or whatever

as for L’H, you can use it to compute a limit whenever the hypotheses are satisfied

there’s no authority that says you can’t use L’H on this or that limit, as long as all the conditions are met

[upbeat yew]

can I use it to compute a limit for a sinus/cosine function?

[prime hornet]

sure, why not?

[upbeat yew]

I try to prove the sinx / x limit
before using the L'H rule

[prime hornet]

well, you’ve got to make sure you have a 0/0 or inf/inf situation

[upbeat yew]

wdym

0/0 is a blank point on graph

inf/inf is infinte in limit

[upbeat yew]

it's undefined for that point

for all x = 0: 0/0 means undefined and we just circle the point on graph

[flint hare]

You can only use L'H on a limit that, when you plug in the value, gives you a 0/0 or inf/inf situation

[prime hornet]

for instance, if you want to compute the limit of x/x as x tends to 0, directly evaluating that function at 0 yields an indeterminate form of 0/0

[flint hare]

A inf/inf limit is not necessarily infinite

[upbeat yew]

this is sinx/x
it's limit is 1

0/0

wait so what's the definition of a limit of a function

[upbeat yew]

and x has to be infinite or x->0?

what does that imply on my first and second derivative of that function?

and how do you prove lim 𝑥→0 sin𝑥/𝑥 = 1
And what does sin(1/x) mean on the limit inside of the sin function

That's why I'm very confused

because for x->k and x->inf
it's something very different

[flint hare]

Are you taking a calculus or analysis class?

[upbeat yew]

yes

exactly

but I'm very confused

[flint hare]

Wdym yes, is it calculus or analysis? Which of those two?

[upbeat yew]

analysis

[flint hare]

Well you should know the definition of limit at this point

[upbeat yew]

say for epsilon delta =0 I can compute the limit for x->0 but that's still confusing

[supple sigil]

it depends on how you define sine

[flint hare]

Usually lim x-> 0 sinx/x is proven with squeeze theorem

There's a classic geometric proof of it

[upbeat yew]

the definition of limit as much as I know is that there is a definition for a sequence and a definition for a function

for a sequence Xn it's limit is if for epsilon > 0 there exists N so that |Xn-x0| < epsilon

x0 = l

And for functions, there are two criterias one by Heine and one by Cauchy:
lim x->x0 f(x) = l if the sequence (xn) is also approaching that limit
and epsilon delta for Cauchy:
0 < |x-x0| < delta => |f(x)-l| < epsilon

x is every point and x0 is a given point on graph which is the limit, correct?

[cerulean oyster]

Without geting too much into the actual definition, the idea is that the two-sided limit of a function exists if near the x-value of interest, the function approaches a specific number from the two sides.

there isnt a need for that specific x to be part of the domain.

[upbeat yew]

wdym part of the dom x?

[supple sigil]

you can define the notion of a limit of a function f at a point x, without requiring that f be defined at x

[upbeat yew]

yes it's the same as with sinx / x
x ≠ 0

[upbeat yew]

but when we use funtion sign(x)
we see that when approaching to 0 from right the function range value is 1
and from left the function range is -1
so there is n't a two-sided limit at this point of dom x

[cerulean oyster]

Yep, thats correct

[upbeat yew]

There is also the definition of e (euler number)
which is used later for limits which is also confusing to me

I still need to understand the definition of limit and how to use the L'H rule

for computing the limit of any function and then the limit of sin(1/x)

[cerulean oyster]

,tex $\big(1+\frac1n\big)^n$ i suppose?

[upbeat yew]

and what does that imply on my first and second derivative

[cerulean oyster]

With tending to infty.

[upbeat yew]

yes exactly that's the e

what's is the definition of the limit and how to use the L'H?

say when lim x- >x0 or lim x->infty
what's the difference?

[cerulean oyster]

The delta-epsilon definition of a limit is a little more involved, but there are videos that explain the idea nicely.

[upbeat yew]

the squeeze theorem uses the limit as well

which is also quite misleading to the definition of limit for me

I'm still confused of the actual usage of L"H - how to use it, and what are criterias of it

[cerulean oyster]

Gimme a sec

[supple sigil]

Since you're in an analysis class, any tools you are expected to use (like L'H) should have been proved beforehand and you should know the precise theorem statement and conditions

[upbeat yew]

yes the L'H have been proved somehow before, but i'm still confused of it

[cerulean oyster]

tbh i never came to give a full read to the proof of L'H

[supple sigil]

if it has been proved then you should know the full statement of it

just check your notes or online

[cerulean oyster]

I can help you with the criteria of use and how to apply, im out of my expertise for the reasons of them

[upbeat yew]

The L'H was proved using the generalized mean value theorem
and finding case study of 0/0

[upbeat yew]

I know that it's mainly used for uncertain situations that's the statement of L'H.
it also defines something for the derivation which is also unclear to me yet

[cerulean oyster]

Also, the definition of e as a useful constant relies on the limit itself, so its just how its defined.
We generally use that identity to solve other limits

Previous to that we used an approximation for logarithms

like, 500 years ago

[upbeat yew]

it's mainly used on log e (ln)

[cerulean oyster]

yep, had to do with economics and interest compound

[upbeat yew]

and x has to approach infty for a limit, when f(x)= e^x

,tex f(x)=e^x

[grand pond]

macwindow

f(x)=e^x
```Compilation error:```! Missing $ inserted.
<inserted text> 
                $
l.49 f(x)=e^
            x
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

LaTeX Font Info:    Trying to load font information for OT1+lmr on input line 4
9.
(/usr/local/texlive/2023/texmf-dist/tex/latex/lm/ot1lmr.fd```

[cerulean oyster]

,tex is the default enviroment

You have to put things inside $

Everything else comes out as text

[upbeat yew]

,tex $f(x)=e^x$

[grand pond]

macwindow

[upbeat yew]

so l"h is very useful

[cerulean oyster]

yeah, somewhat.

[upbeat yew]

it comes very handy when trying to find any limit of a function

but for a sequence no?

I'm still confused of the proof of L'H and limit

[cerulean oyster]

you can use it too.

x -> 0, sin(x) / x is a prime example to use lhopital, but we all know it tends to 1.

[upbeat yew]

so my question is:

how do you proof l'hoptial's rule
usage of it
and proof of sinx /x so I can also understand lim x->k (cos(x)/x)

[cerulean oyster]

i cant recall the proof of LH, mb i guess

The usage of it is pretty easy

I can help with that if you wanna

[upbeat yew]

yea sure

how do I use it

[cerulean oyster]

the general idea is:

,tex given two functions $f(x)$ and $g(x)$, which, for some $c$, under $\displaystyle\lim_{x\to c}$ both tend to $0$ or $\pm \infty$
$$\lim_{x\to c} {f(x) \over g(x)} = \lim_{x\to c} {f'(x) \over g'(x)}$$

[cerulean oyster]

thats the general idea, aka

if you have (or can reach) 0/0 or ∞/∞

[upbeat yew]

,tex the squeeze theorem says:
if:
$for all a≠x ∈(a-epsilon,a+epsilon)$, we have:
$g(x) ≤ f(x) ≤ h(x)$
$If lim x-a g(x)=lim x->a h(x) =L$ then $lim x->a f(x) = L$ as well
So it's basically saying they are all equal

[grand pond]

macwindow
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[cerulean oyster]

Avoid latex, you dont seem to know the markdown yet 🥀

The idea of the squeeze theorem applies both to f(x) and sequences

[upbeat yew]

,tex but how do I find what's c for $lim_{x\to c} f(x)$?

[grand pond]

macwindow

[cerulean oyster]

Depends on the case

[upbeat yew]

is c the domain of the function?

[cerulean oyster]

No, c is some value x approaches to. If said limit exists, then we say f(x) approaches L

[upbeat yew]

,tex $x \neq c$ so I calculate: $\displaystyle \lim_{x\to c} {f(x)}$

[cerulean oyster]

=/= is \neq

[grand pond]

macwindow

[cerulean oyster]

What does this have to do(?

[upbeat yew]

how do I know to where x approaches?

[cerulean oyster]

you just choose, or youre told

[upbeat yew]

and how do I solve it?

the limit for x = 2 is:

?

x = 2 so : sin(2*2π) / 2^2 - 4?

[cerulean oyster]

Theres usually a sort of step-by-step we do to solve, starting from the easiest techniques to hardest

First, just try to evaluate by plugging 2 into x and see what you get

[cerulean oyster]

Mb for bad tex.

[upbeat yew]

,tex $\displaystyle \lim_{x\to 2} {sin(4 \pi) \over 2^2 - 4} = {sin(4 \pi) \over 0}$ ?

[cerulean oyster]

Try to see what is the value of sin(4pi)

[grand pond]

macwindow

[upbeat yew]

,tex $\displaystyle \lim_{x \to 2} {sin{4 \pi} \over 4 - 4} = 0$

[grand pond]

macwindow

[cerulean oyster]

?

[upbeat yew]

it's zero

sin(4π) = 0

0/0 = 0

for lim = 0

[cerulean oyster]

0/0 is undefined as part of a limit

[upbeat yew]

and for sin(1/x)
x = 0 is horizontal asymptote right?

and vertical asymptote (lim) doesn't exist

so now I need to check both sides whether the function is going up or down and plugging x=0 for derivative as non-existent

/undefined

correct?

[cerulean oyster]

I have 0 clue what youre talking

And you probably should start from the basics of limit evaluation

[upbeat yew]

,tex $\displaystyle \lim_{x \to \infty} {sin(1 / x)} = \infty$

[grand pond]

macwindow

[upbeat yew]

and there is asymptote for x=0

[cerulean oyster]

Ok, and prob start dealing with trigonometric functions too.

[upbeat yew]

ok so I need to know how to use l'h and the proof it,
and how to prove sin(x)/x limit = 1 using geometry

[cerulean oyster]

This is wrong

[upbeat yew]

why?

[cerulean oyster]

Do you know how sin x looks?

[upbeat yew]

yes

[cerulean oyster]

if so, any clue why "infinity" cant be the answer

[upbeat yew]

oh

so it's 1

because it can't go higher tahn 1

[cerulean oyster]

Nope.

[upbeat yew]

so?

[cerulean oyster]

undefined

[upbeat yew]

?

is it undefined?

[cerulean oyster]

Graph it out and look at it

Using desmos or geogebra

You will have to go an re-learn limits, probably from a perspective of calculus.

[keen sinew]

? the limit is just 0

am i missing smth in the conversation

[upbeat yew]

can you please define limits, what they mean, how to use l'h, how to calculate them?

that's my question

|x-l| < epsilon

etc

for sequences and functions

[keen sinew]

thats the question you have to answer or the question you are asking ?

[upbeat yew]

I'm asking]

I don't understand it

[keen sinew]

ohh

[upbeat yew]

in analysis

[keen sinew]

so you don't understand the epsilon-delta definition of limits ?

[upbeat yew]

yes

that and 0 < |x-l| < eps
etc

what's x0

etc

[keen sinew]

alright

[upbeat yew]

that's something that's confusing me

[keen sinew]

lets start with sequences

thats easier

[cerulean oyster]

🥀

[keen sinew]

if $u_n$ converges to a limit $l$

[grand pond]

robins

[keen sinew]

then the definition says that for every interval, we can find a rank "$n_0$" from which every $u_n$ is in that interval

[grand pond]

robins

[keen sinew]

So

Let $\epsilon > 0$, we can find $n_0 \in \mathbb N$ where for every $n$ greater than or equal to $n_0$, we have that $u_n$ is in $(l-\epsilon, l + \epsilon)$

[grand pond]

robins

[keen sinew]

so it translates to :
$\forall \epsilon > 0, \exists n_0 \in \mathbb N, \forall n \geq n_0 : |u_n - l| < \epsilon$

[grand pond]

robins

[keen sinew]

that means that you can go as close as you want to the limit from a certain rank

so for functions, you have the same idea, but more complex. Lets say that $\lim_{x \rightarrow a} f(x) = l$ with $a,l \in \mathbb R$

[cerulean oyster]

"\lim" btw

[grand pond]

robins

[keen sinew]

ty

sorry for being a typst user 🙁

[keen sinew]

so now you want to say that : for every interval centered at $l$ , we have an interval centered at $a$ such that for every $x$ in that interval, $f(x)$ is in the interval centered at $l$

[grand pond]

robins

[keen sinew]

now, let $\epsilon$ be the radius of the interval centered at $l$, we affirm that there exists a radius $\delta$ such that for every $x$ in $(a-\delta,a+\delta)$, we have that $f(x)$ is in $(l-\epsilon, l+\epsilon)$

[grand pond]

robins

[keen sinew]

let's say that f is defined on the domain $I$. That translates to :
$\forall \epsilon > 0, \exists \delta > 0, \forall x \in I, |x-a| < \delta \implies |f(x) - l| < \epsilon$

[grand pond]

robins

[keen sinew]

this notation is just a way to highlight that we talk about distances : $|x-y|$ is the distance between $x$ and $y$ in the real line

[grand pond]

robins

[keen sinew]

so limits in the real line are useful tools to understand the behaviour of a function at infinity, or at some value for which our function is not defined.. To compute limits, you have many useful theorems/methods

the most useful ones are the squeeze theorem, the comparative growth theorem, the l'Hôpital's method and some other specific methods

[midnight plank]

@upbeat yew Has your question been resolved?

[keen sinew]

Ive seen that you wanted a proof of l'Hôpital's rule... thats far from being trivial but I can write you a proof

[upbeat yew]

ok

but before the l'h, what do you mean by rank and interval?

[upbeat yew]

also, what's x and x0 when doing epsilon delta?

like interval is this: [] () (closed and open set)

and rank is the position of the element inside the set?

[keen sinew]

you don't have an x0, or it replaces your delta

[keen sinew]

a rank is just a positive integer

[upbeat yew]

not a number for element position inside the set?

[keen sinew]

the position in a sequence

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[upbeat yew]

.reopen

[midnight plank]

✅ Original question: #help-49 message

[upbeat yew]

if you can also prove that l'h rule and the usage, it can be very helpful

[keen sinew]

okay

[upbeat yew]

also is derivative of a function related to the limit of a function?

[keen sinew]

urd

yes