help-4

[worn locust]

is d) 0?

[hazy pivot]

Nope

Once again see the arrow

[worn locust]

negative infinity?

i might be slow

[hazy pivot]

Yes

[hazy pivot]

You're not lol

No one's slow, we are all on our own paces

[worn locust]

I'm glad calculus wasn't on the SAT

I would've failed it

my guess for e) is -4

f) positive infinity

g) -1

h) -4?

oh wait h doesn't exist i think

[worn locust]

are u there

[hazy pivot]

Sorry gimme a moment

Just gonna forward it here so I can check

[worn locust]

bet

[hazy pivot]

f doesn't exist

Cuz the function isn't defined below x = -10

[worn locust]

oh ok

wait

[hazy pivot]

g isn't -1

Like I said, the value at the point doesn't matter

It's the value around the point

[worn locust]

does this not count as the function?

[lyric dagger]

hey calc student here mind if I contribute?

[worn locust]

no problem

[hazy pivot]

Yes but what happens when x < -10

We don't have anything for that part

[hazy pivot]

Yeah sure hop in

[lyric dagger]

what letter are you on?

[hazy pivot]

Currently f, maybe g

[worn locust]

when its -9 its -4.78 right?

[worn locust]

f

[hazy pivot]

-9 isn't less than -10

[worn locust]

oh

right

[lyric dagger]

f would be positive infinity

[worn locust]

thats what i thought

[hazy pivot]

No it wouldn't lol

[worn locust]

but apparently it doesnt exist

[hazy pivot]

The function isn't defined for x < -10

It goes to positive infinity at -10

But we have no information about what happens below

[worn locust]

isn't it assumed that it goes to beyond x<-10 eventually?

[hazy pivot]

That's not what an asymptote is

[lyric dagger]

Oh I see

[hazy pivot]

It's bad practice usually to assume stuff that you aren't given

[worn locust]

but isn't this part beyond x<-10 already

its past the line

[hazy pivot]

Fair enough, it is imprecise

Usually this means for some value -10-a, where a is very small, it goes to infinity

Ask your teacher what their intention was

That's probably the best

[worn locust]

okay

and then g is 3?

[lyric dagger]

yeah

[worn locust]

and h doesn't exist

because its 2 diff values?

[hazy pivot]

Yes

[worn locust]

cool thank you

[lyric dagger]

so i'm not crazy, f would be up to positive infinity if the function didn't cross x=-10?

[hazy pivot]

It's unclear essentially. I thought the arrow indicated it asymptotically going to +inf at a point just below -10

But yes you could interpret it the way you two did

And the best way to be sure is to ask the teacher was they intended

[lyric dagger]

it's hard to say really because there are no exact functions for the lines

[hazy pivot]

Yeah

[lyric dagger]

so you couldn't plug in values and check for exact holes

👍

[worn locust]

.close

[vale dock]

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[gilded summit]

I never know how to approach these common sense proofs, so I was curious if I did this properly:

[gilded summit]

like do i need to show more work or does this make sense?

idk what else i would write

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@gilded summit Has your question been resolved?

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• Show your work, and if possible, explain where you are stuck.
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[craggy dirge]

Need help with all these

[stark wedge]

ok let's see

have you done systems of 2 equations in 2 variables before? @craggy dirge

[craggy dirge]

Ueah

I need to single out 2 rows

[stark wedge]

right so you know such methods as elimination and substitution yeah?

[craggy dirge]

But idk how

[craggy dirge]

Ye

[stark wedge]

yeah these still apply here

let's start with this one

do you want to do it via elimination, via substitution, or let me decide on the method for you?

[craggy dirge]

Elimination

[stark wedge]

ok

it'll be convenient for us to number the equations like so, just for ease of reference

[craggy dirge]

R1 r2 r3?

[stark wedge]

eh if you want

are you planning to write it in matrix form?

[craggy dirge]

Just like (x,y,z,)

[stark wedge]

i mean are you planning to write it like this:

[ 1  1 -1 | -9 ]
[ 2 -3  2 | 13 ]
[ 3 -5 -6 | -15]

or not?

(perfectly fine if not, btw)

(just feeling out your method preferences)

[craggy dirge]

I dont really know how to

[stark wedge]

ok nevermind that then, we'll just keep it algebraic and unshortened.

[craggy dirge]

This was a problem i did before

[stark wedge]

looks alright

at least the process is correct, i can't vouch for the arithmetic as i don't wanna spend time looking over it rn

another choice then: which of the 3 variables do you want to eliminate -- x, y, z, or let me decide?

[craggy dirge]

What ever is easier

[stark wedge]

they're all equally easy in this case, but let's pick z for elimination.

[craggy dirge]

K

[stark wedge]

take equations (1) and (2) from here, and eliminate z from them

do just that, and show me what you get. don't go further yet

[stark wedge]

ok excellent

we can label 4x-y=5 as equation (4) if we wish.

now take the equations (1) and (3), and eliminate z from those too.

same process, and show me your result.

[craggy dirge]

Is that right?

[craggy dirge]

Done

[stark wedge]

ok, looks good

now take equations (4) and (5) [the one you just got] and solve them as a 2×2 system

after that, plug your known x and y into any of the three original equations you want [but i recommend (1) as it's easiest] to get your z back

[craggy dirge]

So

I have this now

Correct?

Multiply the bottom by -11

I got that as my answer

Is that correct!

Nope

Doesnt work when i plug it in

@stark wedge not working

[stark wedge]

lets see

[stark wedge]

these should be good

part of your other stuff seems to be cut off...

[stark wedge]

also your handwriting is tripping me up where some 4's look like u's and 2's look like z's

oh you plugged x=-2 into (5) wrong

you should have had -8 - y = -5

this would give you y = -3

that seems to be your error

[craggy dirge]

Ohhh

So for The next one

I did this

Im gonna scrap that

I wanna isolate z

[stark wedge]

you mean eliminate z?

[craggy dirge]

Yea

[stark wedge]

isolating would mean something different

anyway ok

[craggy dirge]

Eliminate

So

[stark wedge]

r2 & r3 are easy to eliminate z from

[craggy dirge]

Oh yah

[stark wedge]

and then also perhaps r1 & r3

[craggy dirge]

So

This

The multiply the bottom by-6

But it doesn’t work out

@stark wedge i get like 23and 11

Which dont divide nice

[stark wedge]

you mean that your x and y don't work out to be integers?

[craggy dirge]

Yeah

I get 5x+y=2

And 7x+6y=1

[stark wedge]

let me reproduce this myself

7x+6y = 1?

[craggy dirge]

Yeah

[stark wedge]

2*r3 is 6x + 8y - 2z = -4
r1 is x - 6y + 2z = 5

[craggy dirge]

Oh

Thats where i went wrong

[stark wedge]

yeah unfortunately systems of equations are kinda always a minefield for arithmetic errors

[craggy dirge]

X=-1?

No x is 1

Cuz -3x=-3

Can u confirm? @stark wedge

[stark wedge]

i get x=1

[craggy dirge]

One last one i need help for

Number 5

I did 3 4 6 thanks to you

But number 5 has one equation wirhout z

NVM i got it

Omg

Thanks you so much

[vale dock]

@craggy dirge Has your question been resolved?

[vale dock]

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• Show your work, and if possible, explain where you are stuck.
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• Be polite and have a nice day!

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[wild cloak]

Im trying to understand the philosophy of limits

[wild cloak]

so basically limit is about how a function behaves when you approach some value

for example what will be the value of y = f(x^2) when you approach 2

right?

if this is the case does that mean the slope we get is approximation?

becuase he is never 0

[red tulip]

thats just the derivative of x^2

using first principles

[wild cloak]

okay?

but thats not the question

derivative comes from this defination

and it uses the limit

[red tulip]

what happens when you make h zero in this case?

like without simplifying

[wild cloak]

undefined

[red tulip]

yes but we still have a solution

thats what limits are for

[pliant nova]

what even is your question

[wild cloak]

im trying to understand that the value we get from limit is approximation

because you cant get exact slope when h cant be equal to zero

its very close to but exact

[pliant nova]

h isnt 0, thats the point

[wild cloak]

then why we put h = 0?

[pliant nova]

where does we say h = 0? we only have h -> 0...

[wild cloak]

we are putting h = 0 to get the limit

you will get 2x

as derivative

[pliant nova]

you mean the last step?

[wild cloak]

yeah

it will 2x + some small value

not 2x

[pliant nova]

yes which is equivalent to 2x since it's "infinitely small"

[wild cloak]

yeah so its equivalent and not equal right?

[red tulip]

Limits arent exclusively for slopes

Btw

[wild cloak]

i know

its just for asking the question

[pliant nova]

no it's equal

[wild cloak]

how can it be equal?

[pliant nova]

we just say that h is so small it makes no difference

[wild cloak]

when h is not zero

[pliant nova]

you can think of it as h being 0 but that breaks the definition doesnt it

[wild cloak]

you go from idea to defination

not from defination to change your idea

[pliant nova]

what

ok

[wild cloak]

all i want to know is

[pliant nova]

do you know that 0.9 repeating is equal to 1

[wild cloak]

yeah that too

[pliant nova]

its the same idea

[wild cloak]

yeah

so

its also approximation

then cause of

limits

[pliant nova]

no man its infinitely small so it makes no difference

[wild cloak]

yeah but it is there

something is there

its not zero

we are ignoring it

doesnt mean its not there

am i right?

[pliant nova]

yes its there but it makes no difference

that's why at the end we just ignore h

[wild cloak]

yeah so slope are approximation

[pliant nova]

no it isnt

[wild cloak]

or any other value we get from

limits

[wild cloak]

.

[pliant nova]

2x is exactly the derivative of x^2

[wild cloak]

you get derivative by limits

[pliant nova]

we agree that 2x is not an approximation there, right?

[wild cloak]

i dont agree

[pliant nova]

How

[wild cloak]

you get derivative by that slope defination btw

so if i dont agree on h = 0

i dont agree on 2x

its that simple

[pliant nova]

so you want to write

2x + h

every time

[wild cloak]

okay so what you trying to say is just so we dont have to write h everytime

we ignore it?

[pliant nova]

ok look have you even used the limit definition to find the derivative of x^2

plug it in and see how it works

[wild cloak]

i know how to use limits

thats not the point

[pliant nova]

what is your question

[wild cloak]

im not trying to learn how to solve limits

[pliant nova]

i know that

[wild cloak]

i want to know is

are limits approximation?

cause we are approaching something

[red tulip]

Yes

[wild cloak]

we are not getting there exactly

[wild cloak]

source?

[red tulip]

Take the function x^2/x

If you simplify it its just y=x

But at x=0 it should be undefined since you cant divide by 0

But with the simplification you see it does pass through the origin

[wild cloak]

so we basically assume x != 0 just to cancel out x and then we put x = 0

[red tulip]

Yes

[pliant nova]

is your question about limits or the limit definition of a derivative...

[wild cloak]

well this cant be real

[wild cloak]

philosophy of limits

[wild cloak]

if thats the case

[pliant nova]

[wild cloak]

integration is also approximation?

casue we use rectangles

[pliant nova]

mate the idea is that the approximation is infinitely close so its basically exact

[red tulip]

Yeah integration is a) the opposite of a derivative b) the division into rectangles as you said

[wild cloak]

no exact is not same as infinitely close

its approximation

[pliant nova]

oh my god

im not here to play socrates

[wild cloak]

well good for you

im clear about my question

[pliant nova]

sure you can keep saying its not exact but idk what you gain from this

[red tulip]

Did you not pick up the word 'basically'

[wild cloak]

im trying to understand if im right or wrong

[red tulip]

Fall is doing their best

[pliant nova]

you are both right and wrong

really tickling your philosophy side there

[wild cloak]

0.33333... = 1/3

so

how i can be right?

[lyric sundial]

??

[red tulip]

Ig limits arent rly an approximation because you are finding the precise value it approaches

[wild cloak]

1/2 + 1/4 + ... = 1

[lyric sundial]

Yes, so????

[pliant nova]

hes saying its not. basically.

[wild cloak]

well let me get the defination

[lyric sundial]

You use limits to find the exact value a function approaches to

[lyric sundial]

@wild cloak

[red tulip]

The limit is the operator that returns the value a function approaches towards a certain x value

[pliant nova]

"how can it be exact if it isnt exact" is basically their argument...

[lyric sundial]

Yeah indeed

[wild cloak]

epsilon is > 0

[lyric sundial]

Sure

[wild cloak]

so they are not exact same right?

[lyric sundial]

.....

[wild cloak]

the sum im getting from limits

is not exact

[pliant nova]

💀

[red tulip]

Im so confused what ur saying ngl

[lyric sundial]

I think you didn't understand what limits are

[pliant nova]

hes saying its not exact because of the infinitely small value

[wild cloak]

yeah

[red tulip]

0.99999999999999 != 1 ahh person

[pliant nova]

bud its infinitely small 😭

[lyric sundial]

Limits don't deal with the x0 you're approaching

[wild cloak]

.....

[lyric sundial]

It only looks for neighbourhoods

[wild cloak]

okay sure then for example if we find slope

we only looking at neighbourhood

[lyric sundial]

Forget the derivatives for now

[wild cloak]

not exact point

[lyric sundial]

You first have to understand limits

[red tulip]

@wild cloak if you walk to a wall you have to reach the halfway point to the wall, then the halfway point from there, and so on for infinity but you can still go and touch the wall

[wild cloak]

obv because i wont to able to do movements of 0.0000000000000x

[wild cloak]

i understand the defination of limits

[lyric sundial]

Mmmhg

[wild cloak]

i can do the problems of limits

its not about that

all i am asking is

if we are only approaching

and not going on the exact point

why we use limits from getting the exact value

for example that 0.99999.... = 1 case

limit will approach a really large number

okay

[pliant nova]

dont think you get what "infinity" means

[wild cloak]

thats cool

[wild cloak]

okay maybe

can you explain me please

[pliant nova]

infinity isnt just a large number

like sure. imagine 10^100^100

that's pretty big

[red tulip]

Even if we say they arent equal whats the significance of them not being equal

[pliant nova]

but infinity is endless

when you approach infinity it never ends

[wild cloak]

then 0.99... != 1

[pliant nova]

thats why it's exact

[red tulip]

Its infinitely small, it doesnt matter if you scope in to the size of an atom its still infinitely small in comparison

[wild cloak]

well this cant be true

[lyric sundial]

Limits are defined as (and therefore exactly) the value you approach as you are approximately close to another number. So for example, the limit as x approaches 2 for f(x) = 3x is exactly 6. However, that limit does not care or consider what f(2) is. We could say f(2) = 17 and the limit would still be exactly 6. It just considers what's close to f(2).

[wild cloak]

this is not physics

[red tulip]

Funny because limits are a pretty big thing in physics

[wild cloak]

i refuse to believe mathematicians just ignoring that small part

[red tulip]

Bro

[pliant nova]

well they do. get used to it

[red tulip]

If you were the size of an atom it would still be infinitely small

[wild cloak]

okay

[lyric sundial]

The fact is there isn't a part smaller than the other

[red tulip]

If we consider the mass of an electron negligible then infinitely small is beyond that and is precise

[lyric sundial]

Because for each "small" positive quantity you can always find a "smaller" one

[wild cloak]

@lyric sundial so we are only getting whats close to n = inf right?

[lyric sundial]

Of course

[wild cloak]

so why are we commenting on exact value?

[lyric sundial]

Because I answered your original question

[sharp whale]

the moment I saw "exact value" and "limit" I immediately knew this was a 0.99999... question

[wild cloak]

is there anything that implies

because of limits i got the exact value

[pliant nova]

ur question is really just "why is 0.999... = 1"

[wild cloak]

nope

[lyric sundial]

Wdym?

Sure it is

Why not?

[wild cloak]

?

proof of burden is not on me?

[pliant nova]

yes it is

[wild cloak]

there is dragon invisible dragon in my room? prove its not there

[lyric sundial]

You think limits are not exact but then you agree that 0.9999... = 1?

[sharp whale]

@wild cloak limits are defined like this:

you choose a particular range that you consider small

then a limit is a promise that, if you reach far enough, you will always stay within this range

for example, if you consider a small distance to be 0.00001,

[wild cloak]

im confused that why im asking

why people are sayuing its equal then

[red tulip]

It is equal because its not an approximation

[sharp whale]

that the limit of 0.9, 0.99, 0.999, ... is 1 tells you that you can always add enough 9s to permanently remain within that small distance of 1

hopefully you were paying attention to that i i e

[wild cloak]

....

[wild cloak]

i know that

i know the epsilon delta definition

[sharp whale]

now consider a continuous function that acts just like 0.9, 0.99, 0.999, ...

1 - 1/10^x

for x = 1, this is 0.9
for x = 2, this is 0.99
for x = 3, this is 0.999

[wild cloak]

okayyyy

so its continuous function

[sharp whale]

we are working with real numbers

now real numbers tells us something neat

[red tulip]

Finding limits involves simplifying the expression and then inserting a value, thats no approximation man

[sharp whale]

for real numbers, 0.99999... = 1 for this reason

[wild cloak]

ok @sharp whale i understand where you are going

but your assumption is

function is continuous

[sharp whale]

hold on there i i e

[red tulip]

O.99999999 = 1 comes from limits

[sharp whale]

you dont know where Im going with this

@wild cloak are you familiar with the monotone convergence theorem?

[wild cloak]

no

[sharp whale]

so much for understanding where I was going

you are already aware that $\lim_{x\to\infty}1-\frac1{10^x}=1$

[rocky lotus]

mtt

[wild cloak]

nope

this is my question

[sharp whale]

i i e please listen

[pliant nova]

he doesnt accept that limits are exact and you go with that😭

[wild cloak]

yeah

[lyric sundial]

You said you KNEW how to solve limits!!!

[wild cloak]

well i can memorize

[sharp whale]

I didnt even say that the limits are exact

wdym you dont believe its 1

this is just a continuous function, you can see the x there

[lyric sundial]

Limits are not things to memorize?!?!?

[sharp whale]

I did not yet connect this to the 0.99999... thing

[wild cloak]

i can memorize oh n = inf, and power in denominator

[sharp whale]

consider it completely separate

[wild cloak]

oh large number

[sharp whale]

i i e DO NOT DO THAT

[wild cloak]

its zero

[sharp whale]

cmon man

[sharp whale]

no

[wild cloak]

well i dont agree with your assumption

why are you still going 😭

[sharp whale]

you didnt even hear me out completely

[red tulip]

Are you saying you dont trust the cars spedometer when you are accelerating

Because thats pretty much a limit

[sharp whale]

you just assumed you knew where I was going then said it was stupid

thats probably because your assumption of my work didnt pan out

[wild cloak]

that doenst prove limits are exact

or approximation

[sharp whale]

the reason you were taught that is because its an easy shortcut to evaluate, that is not the proper version of how limits are supposed to be evaluated

there is a view of limits that does not require exactness

[lyric sundial]

There's nothing to prove. It's the structure of the limit definition that tells it's an exact value

[sharp whale]

you dont seem intent on ever hearing about it

[wild cloak]

okay go ahead

listening

[sharp whale]

got your attention a second time already in 5 minutes

[red tulip]

The value of the function APPROACHES the value

[sharp whale]

dont interrupt ok?

[red tulip]

That doesnt have to have an infinitely small approximation its just true

[sharp whale]

lets first look closely at the epsilon-delta definition

it requires that "there exists an N where |f(x) - L| < epsilon for x > N"

now notice here nothing is being exact

instead of f(x) being equal to L,

we have f(x) being some epsilon from L

the limit only promises that you can get as close as you want

and this is what we consider to be good enough

0.999999... = 1, not because they necessarily represent the same number (which you dont believe),

but at least because adding more 9s always gets closer to 1

and so the limit of 0.99999... is considered to be 1

does not need to match 0.99999...'s actual infinite behavior, if you consider such to be different

you cannot lie about the limit, it does not ask for much, and it says 1

what you can lie about are the axioms of real numbers

we then use a few other properties of real numbers to show that 0.999999... must equal this limit of 1

and leave you no room but to either accept this, or that you dont belong in the real numbers

[wild cloak]

well thats not true

[red tulip]

If you multiply 0.9bar by 2 you get an infinitely close value to 2 and the same for all multiples

[lyric sundial]

?? It IS true

[sharp whale]

then prove it to me that its false

can you show me an epsilon?

[wild cloak]

well

[wild cloak]

.

epsilon has to be greater than zero

[sharp whale]

and?

[wild cloak]

its not zero

[lyric sundial]

So??

[sharp whale]

I never said its zero buddy pal

[wild cloak]

so you cant be "as close you want"

[sharp whale]

i i e, tell me

consider the set of all positive numbers

this set does not include 0

[sharp whale]

what is the smallest number in this set?

[wild cloak]

idk

[sharp whale]

tell me i i e

tell me the smallest number

can you make one?

[wild cloak]

well

you cant

[sharp whale]

"as close as you want" never means "exactly close"

only "nonzero distance close"

"as close, but never exact, as you want"

and by the promise of there never being a smallest positive number, our epsilon-delta definition still holds weight

[wild cloak]

yeah this is true (according to defination"

[sharp whale]

there you go

so dont say "exact" okay?

youre going crazy

you keep throwing this word into a place that the limit ITSELF does not include

the limit does not include "exact," only "close," understood?

[wild cloak]

yeah

[sharp whale]

lets now bring something important

[wild cloak]

i already understand that

[sharp whale]

no you didnt

you misread my words to be something else

now if you did understand this, why twist them?

[wild cloak]

huh?

[sharp whale]

you already knew the limit cannot ever have epsilon = 0

why did you

[wild cloak]

im not being rude but are you in college?

[sharp whale]

Im not being rude, but were you born yesterday?

[wild cloak]

huh?

[sharp whale]

you cant get away with that kind of goalpost shifting

understood?

[wild cloak]

what??

[sharp whale]

good

lets move on to something of actual importance

I havent done this in a while, so we'll need to dust off a few axioms about real numbers

[wild cloak]

what is blud saying 😭

[sharp whale]

now by axioms, I mean things that are so true that they might as well be given

they may not be the actual axioms, but theyre properties we have to accept

youll need these properties for what we'll say next to hold weight

[wild cloak]

okay?

[sharp whale]

first: https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

[wild cloak]

are you trying to prove the 0.99... = 1

[sharp whale]

real numebrs obey a property called "completeness"

[wild cloak]

cause i already know this proof

[sharp whale]

tell me the proof then

[wild cloak]

do you think i haven't researched about this?

before asking here

[sharp whale]

you didnt seem to know what a limit was

[wild cloak]

?

[sharp whale]

given you tried to say epsilon = 0 more than zero times

epsilon = 0 is not within the jurisdiction of a limit

[wild cloak]

my bad bro im too dumb to understand you

did you even see the original question ?

btw i didnt ask about 0.99... = 1 proof

it was not even the question

[sharp whale]

yea, it says upfront you didnt get the philosophy

[wild cloak]

yeah thats why im asking it here?

[sharp whale]

itd be pretty important

if as part of that philosophy,

you need to now accept that 0.9999... = 1 as an inevitable consequence

[wild cloak]

?

what?

[sharp whale]

put it this way

how can you use a philosophy if you know a case where it doesnt work?

limits by themselves do not have the power to convey exactness

[wild cloak]

....

[sharp whale]

but combined with the real numbers, and continuous functions, they can

[wild cloak]

"continuous functions"

okay how do you know if a function is continuous or not

can you please tell me

[sharp whale]

so much for that research you were doing...

there are a few separate definitions depending on the axioms you accept are true

[wild cloak]

why are you offended 😭

what

[wild cloak]

.

[sharp whale]

I mean you did say this earlier

[sharp whale]

and then

its the same kind of question Im asking you, you can chill on that

[wild cloak]

knowing something doesnt mean i agree with something

bruh

[sharp whale]

during your research, did you look into topological spaces?

[wild cloak]

no?

why would i

[sharp whale]

for metric spaces, we can use a particular definition of continuous that can help us out a whole lot

[wild cloak]

okay i agree with everything you said

but can you tell me

[sharp whale]

stop interrupting me telling you the definition

[wild cloak]

how to check if function in continuous or not

[sharp whale]

a continuous function is a function that commutes with limits

in other words,

[wild cloak]

....

[sharp whale]

if you had a sequence of points that converged to a value,

x1, x2, x3, x4, x5, ... -> x,

then a continuous function lets you conclude
f(x1), f(x2), f(x3), f(x4), f(x5), ... -> f(x)

doesnt matter what sequence you pick

as long as it converges, this will then also follow

[wild cloak]

..

i know this

[sharp whale]

good

this definition can be proven to be equivalent to the epsilon-delta definition

[wild cloak]

but how do you check if function is continuous or not

[sharp whale]

you can check if its continuous using a few methods

[sharp whale]

the first method is pretty direct, its not pretty but it works

the second method is to use some properties that continuous functions have

if f and g are continuous, then their sum is also continuous

[wild cloak]

.....

[sharp whale]

then their difference, product, and quotient (assuming g is nonzero) is also continuous

then their composition (assuming certain areas are also continuous) is also continuous

with this, you can create a set of continuous functions

[sharp whale]

i i e

buddy

if we used that

youd conclude 0.999999... = 1

so far it sounds a whole lot like you cant understand everyone else when theyre telling you this

[sharp whale]

this doesnt even work when a is infinite

we cant even use that

[wild cloak]

yeah

thats what im saying

[sharp whale]

then why did you mention something that didnt work to me, who's not talking about it?

please dont waste our time

[sharp whale]

do you want me to use this?

[wild cloak]

well the definition itself use limits

so

[sharp whale]

definition

[wild cloak]

no point

[sharp whale]

that doesnt tell me anything

its a yes or no question

no point implies "maybe"

uses, not use

[wild cloak]

dont use it

[sharp whale]

the way that wouldve been used wouldve been longwinded anyway

[wild cloak]

...

nvm you wont understand what im i asking

its my fault

[sharp whale]

you can try asking it, just to be sure

we've already covered what limits by themselves do

[wild cloak]

give me some time i gotta take a break

[sharp whale]

@wild cloak btw if youre asking about how a limit can define the derivative to line up with the usual notions of "slope at a point," I dont have a completely solid reason but I do have one thats solid enough that you can hear out

[sharp whale]

there is also a bit of a bitter pill here: the exact value never exists, we just use its limit because it acts so like the value we want that we might as well use it

[sharp whale]

so far you are working with mostly continuous functions, these (at least by this principle or more likely by epsilon-delta) mean that the functions you learn in school will not care about extremely small changes in x except at easily predictable moments

[vale dock]

@wild cloak Has your question been resolved?

Channel closed

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[sharp whale]

its such a shame that we couldnt get this ironed out

[vale dock]

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[craggy girder]

need help with this

[sharp whale]

project euler?

@craggy girder also, whats your progress so far?

[craggy girder]

yes

i wrote down all the numbers that have their powers under 100

[sharp whale]

what, on paper?

[craggy girder]

yes

[sharp whale]

youre not supposed to solve these problems manually

[craggy girder]

oh lmao

[sharp whale]

you do realize project euler is written to be solved with computers right

some of the later ones dont seem possible unless by brute force

damn I lost the code that I used to solve the first 40 of these or so

but one of them can in fact be solved by hand

[craggy girder]

this is prob number 29

[sharp whale]

its number 29

that one cannot be solved by hand

[craggy girder]

i had solved #24 by hand too

[sharp whale]

that ones an easy one to solve, yea

[craggy girder]

its simple pnc

[sharp whale]

same with 41

I recommend computers simply because the brute-force or the lists of primes youll need will force you to use one

theres another one that isnt very easy to solve but can be done

it takes several steps and can in fact be done without a calculator

[craggy girder]

nice

but back to this q

2 -> 99
4 -> 50
8 -> 66
16 -> 50
32 -> 80
64 -> 66
 = 411

i wrote this for 2 family

[sharp whale]

are you measuring how many of these arent already covered by a number above?

[craggy girder]

yes

how many unique powers will be there

[sharp whale]

have you already gone through all of the primes?

[craggy girder]

yes

[sharp whale]

what about 6?

[craggy girder]

3 -> 99
9 -> 50
27 -> 66
81 -> 50
 = 265

5 -> 99
25 -> 50
 = 149

[craggy girder]

the same for 6, 7, 10

[sharp whale]

this is actually a lot more doable than I thought

[craggy girder]

yeah

this is it

ther rest will have 99

[sharp whale]

have you tried out the answer?

[craggy girder]

yes its incorrect

[sharp whale]

that sucks

[craggy girder]

Total = 411 + 265 + 4*(149) + (99-18)*99
      = 9291

[sharp whale]

oh wait mb I misread something

this implies you go from 2^2 to 2^100

and also from 3^2 to 3^100

and so on

[craggy girder]

yeah

[sharp whale]

you only went up to 2^6

[craggy girder]

3^2, 3^3, 3^4, .... 3^100

becuase 2^7 = 128 is grt than 100

so we dont have to do 128^1, 128^2 ...

[sharp whale]

that doesnt count, its not asking for all the numbers < 100

it didnt say a^b < 100

look at the example

it includes 5^5, which exceeds a thousand

[craggy girder]

2 -> 99
this means that i take 2^2, 2^3 ... 2^100

[sharp whale]

why not 2 -> 100 then?

[craggy girder]

typo

[sharp whale]

2^(2 ... 100)

[craggy girder]

2^1 not included

[sharp whale]

didnt say that

4^100 is also included

but its missing from your count

[craggy girder]

a in [2, 100]
b in [2, 100]

[sharp whale]

oh mb I completely screwed up

[craggy girder]

becuase i included 2^(2 ...100)

i indirectly included 4^(1 .. 50)

so i have to take addtional 4^(51 .. 100)

which is 50 numbers

hance 4 -> 50

@sharp whale ?

[sharp whale]

Im going through them with a computer to make sure we didnt miss anything

[craggy girder]

oh thanks

s = set()

for a in range(2, 101):
    curr = a
    for _ in range(2, 101):
        curr *= a
        s.add(curr)
        
print(len(s))

this runs instantly 😭

i thought it would take too long so i used maths

[sharp whale]

thats the classic beginning project euler experience

computers run through millions of possibilities extremely quickly

often you can find a few shortcuts that can cut the time in half or so

but youre still in the beginning stages where they dont exepct you to even program your way out of it

later on the problems get harder and then you focus on the computer solution

[sharp whale]

btw:
2 -> 99
4 -> 50
8 -> 50
16 -> 41
32 -> 51
64 -> 37
= 328

[craggy girder]

yes makes sense

4^99 = 8^66

thanks

[craggy girder]

btw how can this be further optimised?

[sharp whale]

well for one, set is already optimized enough to look for duplicates when you insert them in

[craggy girder]

to remove duplicates is the only reason i use sets lmao

thanks again

[sharp whale]

np

[craggy girder]

.close

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[vale dock]

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[paper condor]

Dont mind the unfamiliar language this is a rhombus and AB IS 50cm. Can someone help me find the Length of AC and BD

[paper condor]

Can you use trigonometry to solve this

<@&286206848099549185>

[errant gyro]

!15m i think

[vale dock]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[paper condor]

Can you explain a bi

[ashen prawn]

,rotate

[ashen prawn]

kannada language?

i can comprehend a bit

[paper condor]

Its sinhala

[ashen prawn]

oh

almost similar

tamil derivatives

[paper condor]

Yup

So this is a rhombus

[ashen prawn]

so first check out the properties of a rhombus

then think about your problem at hand

maybe you might get some ideas

[paper condor]

Ik them

[paper condor]

Tho i was doing that for the last hour

[ashen prawn]

dont spare much time for it, try to research the internet

maybe you might encounter similar questions

[paper condor]

Do you know how to solve it?

[ashen prawn]

i dont remember myself the properties

once i recollect them i will be able to do this

[paper condor]

I think i should use trigonometry here

Dunno i cant find a lead

[ashen prawn]

wait then

ill see it after my lunch

mb for the delays

[paper condor]

thank you so much

[main cairn]

I have reviewed the question.

Remember perpendicular opp angles are equal

It might be possible to use trig, im not sure, yet. Ill look into it.

This is a tickey one haha, its been a while since ive done a quesion like this.

alright @paper condor I think Ive figured it out.

[paper condor]

really?

[main cairn]

Each half of a diagonal and side form a right triangle. Correct?

If diagonals are AC and BD

[paper condor]

yes

[main cairn]

then $AB^2 = (frac{AC}{2})^2+(frac{BD}{2})^2$

[rocky lotus]

Basically Macro

[main cairn]

...

bare with me 💀

[paper condor]

no prob

$AB^2 = (frac{AC}{2})^2+(frac{BD}{2})^2$

[rocky lotus]

Jenul

[main cairn]

$AB^2 = (\frac{AC}{2})^2+(\frac{BD}{2})^2$

[rocky lotus]

Basically Macro

[main cairn]

There we go

[paper condor]

yeah

[main cairn]

because AB= 50cm

[paper condor]

yes i see, but even if i put AB^2=50^2 to this equation, AC and BD are still unknown

[main cairn]

you spotted the key issue

[main cairn]

from this we get.........

$50^2=\frac{AC^2}{4}+\frac{BD^2}{4}

...

$50^2=\frac{AC^2}{4}+\frac{BD^2}{4}$

[rocky lotus]

Basically Macro

[main cairn]

$AC^2+BD^2=10000$

[rocky lotus]

Basically Macro

[main cairn]

we need an extra condition to find them individually

ahh @paper condor

we can use your angle condition

we need to simplify $∠CAB+90∘+∠DBA=180$

[rocky lotus]

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

[paper condor]

we can find the square root of them cant we? AC + BD = 100

[main cairn]

correct

okay so look at the right angle triangles in AOB where the diagonals intersect O

[rocky lotus]

Basically Macro

Basically Macro

Basically Macro

[main cairn]

angle CAB is at vertex A between side AB and diagonal AC.

[paper condor]

yes

[main cairn]

Alright so using triangle AOB we get:

[rocky lotus]

Basically Macro

[main cairn]

Since angle DBA is at vertex B between sides AB and diagonal BD in triangle AOB

[rocky lotus]

Basically Macro

[main cairn]

@paper condor do you see what I am aiming for?

[paper condor]

one sec let me think

nope i dont think im seeing what you are thinking

could you give me a lil hint

[main cairn]

pythag

we had $AC^2+BD^2=10000$

[rocky lotus]

Basically Macro

[main cairn]

but thats only one equation for the two unknowns

we needed a second relation betweed AC and BD to solve

as the extra condition in the problem was stated by you in the notes it shows how AC and BD can relate and with out it, there can be many different pairs

[paper condor]

please continue

[main cairn]

we know $cos(Angle CAB) =sin(Angle DBA)$

[rocky lotus]

Basically Macro

[main cairn]

We use the sum condition and sub

soryr

sorry

sub $\frac{AC}{2\times50} = \frac{BD/2}{50}$

[rocky lotus]

Basically Macro

[main cairn]

@paper condor are you able to continue from here?

[paper condor]

oh yeah

thanks a ton bro

your name?

[main cairn]

Basically Macro

[paper condor]

is that your real name?

[main cairn]

lachlan

[paper condor]

thank you lachlan, for taking the time to help me

[main cairn]

No problem.

Is there anything else you need assistance with?

[paper condor]

nope

[main cairn]

Can you also provide me feed back so that I can improve my abilities.

I know I need to work on my typing speed haha.

[paper condor]

what do you mean by feedback? like what you did wrong or right?

[main cairn]

yes

[paper condor]

oh man, I dont know what to say. yeah except for your typing speed everything was great

[main cairn]

Alright, I wish the best of luck!

[paper condor]

thanks, have a great day

.close

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[dark echo]

I don’t need to solve for a value specifically, but just confused on what exactly I look for here?

[stiff lily]

draw tangent lines at those places and sort the slopes

[dark echo]

0 is just there as 0 right?
So g’(0) 0 g’(4) g’(2) g’(-2)
Least to greatest

[stark wedge]

you may wanna put < signs better these for better readability

[dark echo]

Will do next time mb

[stark wedge]

no, do it this time too.

g'(0) < 0 < g'(4) < g'(2) < g'(-2) is your claim, yes?

[dark echo]

Yes

[stark wedge]

.. yeah checks out actually

[dark echo]

Ok perfect thank you

.close

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[rocky imp]

I don't understand how to solve it, I tried asking chatgpt and photomath for help but I am too stupid to understand their explanations

[exotic whale]

ok so basicly

cos(4pi-a)=cos(a) and tan(11pi -a)=-tan(a) same thing for sin

[rocky imp]

but why do the 4pi and 11pi just disappear, does it happen in every single equation of this sort?

[stark wedge]

!nogpt

[vale dock]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[stark wedge]

so the key thing to realize here is that sin and friends are what's called periodic functions.

[exotic whale]

well i think its best is you go back to basic trygonometry that will explain that more. But as a short answer, for cos and sin imagine a circle, let teta be an angle , every 2pi is a full turn so therefore it has the same place in the circle and cos and sin are the same

[rocky imp]

my bad, is just that I don't really have anyone else to ask for help

[stark wedge]

you've seen the unit circle or trigonometric circle before, right

[rocky imp]

yeah

.close

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[exotic whale]

guys is the limit outside of a sigma or outside of that simbole of multiplication the same as the limit inside of it?

[pine prairie]

It depends

You need some sort of convergence theorem to pull it out of a summation

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[quick wagon]

Hi i need help to clarify if i did this correctly, i forgot the lesson cause i was busy most of the time and i cant recall how i even solved it

[quick wagon]

it something after the table of distribution

[vale dock]

@quick wagon Has your question been resolved?

[main cairn]

you did everything correctly including the freq col totals, used the correct formulas an identified median and modal classes correctly, you also used proper sub and solving!

Let me know if you need any explination or further assitance

[vale dock]

@quick wagon Has your question been resolved?

[quick wagon]

My prof didn't tell much about it so i just based off my classmate hw of it and listen to their discussions of it which is unreliable on most of the fact that we all got diffirent answers., but same table

but eh I'll take that

[vale dock]

@quick wagon Has your question been resolved?

[obsidian crown]

how do you solve this? i know you need the absolute value first but the other steps idk them

[hot tide]

#❓how-to-get-help

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[novel ravine]

Hello! This isnt really academical but im trying to solve a puzzle and im not sure how to know if I have all the border pieces... it's a 1000 pieces puzzle, 68.0 x 49.0 in size. The pieces are all the same lenght. I tried looking on internet for a thing i could do myself but i didnt even find how to do it so... if anyone wants to help, thank you! (And tell me how to, i have lots of problems with jigsaw puzzles and that would be of great help to know that)

[verbal hound]

lemme give you basic idea

Total Pieces 1000

Finished Puzzle Dimensions: 68 units (let's assume inches) long by 49 units high.

Row x Columns = 1000

for the grid dimensions ,
aspect ratio , 68/49 =

now do factors of 1000
25 x 40
20 x 50
bla bla

68/C = 49/R
C and R are column and row numbers
R/C = 49/68

,calc 49/68

[rocky lotus]

Result:

0.72058823529412

[verbal hound]

,calc 68/49

[rocky lotus]

Result:

1.3877551020408

[verbal hound]

C= (68/49)R

R C = 1000

,w (68/49) R^2 = 1000 find R

[rocky lotus]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by asdpradosh

[verbal hound]

,calc 25/40

[rocky lotus]

Result:

0.625

Result:

0.4

[novel ravine]

O_O

[verbal hound]

look 25/40 is the closest one to the 0.72

if we take any other no of row and columns , i mean if we fix any of the either row o column itll not give you the natural (counting ) numbe

so 25 rows and 40 columns are good

[lost marlin]

?

[verbal hound]

noo

wait

,calc 35 * 29

[rocky lotus]

Result:

1015

[verbal hound]

,calc 35/29

[rocky lotus]

Result:

1.2068965517241

[verbal hound]

@novel ravine come here

[novel ravine]

Im right here mate

[verbal hound]

you need to find two values such that there ratio is closest to 68/49

which will be your row and column numbers

[lost marlin]

The problem with puzzles is that sometimes they may have a different number of pieces, the error can range upto like ± 20 the advertised number

[verbal hound]

suppose that you got a nad b

then inner rows will be a-2 and inner columns b-2
total inner pieces (a-2)(b-2)

total pieces = a x b

then border picese will be axb - (a-2)(b-2)

id say check with numbers 27 and 37

[novel ravine]

The pieces arent bigger than others in mine

They're kinda all the same length

[lost marlin]

No i meant the number of pieces might be slightly different from what is advertised

It can happen

[novel ravine]

Omfg....

[lost marlin]

So we might need to take this into account

[verbal hound]

,calc 68/49

[rocky lotus]

Result:

1.3877551020408

[novel ravine]

Then how do i know if i'm missing some!

[verbal hound]

,calc 37/27

[rocky lotus]

Result:

1.3703703703704

[novel ravine]

I've been at it for a week

[verbal hound]

,calc 27*37

[novel ravine]

Just for the border

[rocky lotus]

Result:

999

[lost marlin]

I am not saying that is it guaranteed to happen

But it can

[verbal hound]

,calc 25*35

[rocky lotus]

Result:

875

[verbal hound]

,calc 999-875

[rocky lotus]

Result:

124

[lost marlin]

Especially when the puzzle has such a large number of pieces

[verbal hound]

ans is 124 ?

[novel ravine]

Uhhhhhh

Well usually it's not LESS than 1000 pieces right m

?*

[lost marlin]

Nevermind

[novel ravine]

I can count them and keep you updated if you want

[lost marlin]

No need for that

That will be a huge time investment

[verbal hound]

not really

[novel ravine]

Puzzles are already a huge time investment

[lost marlin]

I checked online and chances are that it can have 1008 pieces

[novel ravine]

I have exactly

124 pieces

So i guess im missing none...?

[novel ravine]

Except if it's actually 1008 pieces

[lost marlin]

Uh then good luck i guess

If you have exactly 124

Then chances are none are missing

[novel ravine]

.........

Thank you a lot you two

[lost marlin]

I didn't really do anything

[novel ravine]

I'll just destroy it and do it all over again...

[lost marlin]

It was mainly the other person

They did all the work