is the answer c?

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[cerulean saddle]

Explain.

[solemn mesa]

like explain why i think the answer is c?

[cerulean saddle]

Yes.

[solemn mesa]

if the limit equals 2 then that means theres a hole because a limit can equal something if theres a hole but f(5) wouldnt be able to equal 2 i think idk

[cerulean saddle]

Why not?

[solemn mesa]

because it would have to equal a definite point not a hole when its a hole its dne

[cerulean saddle]

...okay, this reasoning is all over the place.

[solemn mesa]

LOL

hey man im trying

[cerulean saddle]

Look. What is the definition of continuity?

[solemn mesa]

well something thats like

continuous

[cerulean saddle]

No.

[solemn mesa]

oh

[cerulean saddle]

Not "something that's like".

We use precise definitions in math.

[solemn mesa]

ok youre right

a function that's continuous is a function that doesn't have an interruption

[cerulean saddle]

That's still not correct.

[solemn mesa]

ok great

ummm

[cerulean saddle]

I can just tell you.

[solemn mesa]

ok tell me techie literate

yk looking at the other posts there seems to be a concise discussion of question asked --> answer given but im happy that youre giving me a very proactive lesson techie literate

[cerulean saddle]

A function $f$ is continuous at the point $x = a$ if and only if $\lim_{x \to a}f(x) = f(a)$.

[midnight finch]

Techie Literate

[solemn mesa]

ok great

so like would the answer be c

[cerulean saddle]

You tell me. According to that definition of continuity, what must be true if a function is discontinuous at a point?

[solemn mesa]

its like not continuous

[cerulean saddle]

...and when is it continuous?

[solemn mesa]

its

continuous

[cerulean saddle]

According to the definition I just gave, when is a function continuous?

[solemn mesa]

wait

so the answer is c

omg youre so great techie literate

omg omg

ill be back with more later god youre so great techie literate

<3

[cerulean saddle]

I don't want to tell you what the answer is, I want to help you learn why it's the answer so you don't have to ask what ends up being the same question twice.

[solemn mesa]

youre right

[cerulean saddle]

But just to fully explicitly lay out the chain of logic, a function is continuous at a point if and only if the limit of the function as it approaches that point equals the value of the function at the point. Therefore, given that $f$ is not continuous at $x = 5$, and that $\lim_{x \to 5} f(x) = 2$, by definition if $f(5) = 2$ it would be continuous at $x = 5$. Since we are given that it's not, it must be the case that $f(5) \neq 2$.

[midnight finch]

Techie Literate

[solemn mesa]

thank you techie literate

can we go for a double whammy here

i did this one already

but why is it a

[cerulean saddle]

Have you learned L'Hopital's rule yet?

[solemn mesa]

yes

[cerulean saddle]

Just use that. All of these functions at the limit are inf/inf indeterminate forms.

[solemn mesa]

OH

so the first one would be infinity and the third one would be 1

why would the second one be 0 though

[cerulean saddle]

Because what's the derivative of e^x?

[solemn mesa]

e^x

[cerulean saddle]

And what's the derivative of a general polynomial?

[solemn mesa]

is the general polynomial the x^2

[cerulean saddle]

...no.

[solemn mesa]

LOL

[cerulean saddle]

A general polynomial is any function of a polynomial form.

[solemn mesa]

techie literate im gonna keep it raw with you

you saw me explain what a continuity was

please dumb it down

[cerulean saddle]

A polynomial is a function of the form $\sum_{i = 0}^n a_i x^i$

[midnight finch]

Techie Literate

[solemn mesa]

um

ive seen that before

[cerulean saddle]

$a_0 + a_1 x + a_2 x^2 + ... + a_n x^n$

[midnight finch]

Techie Literate

[solemn mesa]

ok so

[cerulean saddle]

So that thing's derivative is what?

[solemn mesa]

im giggiling irl rn

i litewally have no clue

is it not the x^2 + 4

[cerulean saddle]

What if we just had $a x^n$?

[midnight finch]

Techie Literate

[cerulean saddle]

What is the derivative of that?

[solemn mesa]

a . n x^(n-1)

RIGHT? RIGHT?

[cerulean saddle]

^(n - 1)

And the derivative of a sum is the sum of the derivatives, right?

[solemn mesa]

yeah....

[cerulean saddle]

Therefore the derivative of $a_0 + a_1x + a_2x^2 + ... + a_nx^n$ is?

[midnight finch]

Techie Literate

[solemn mesa]

ummm its like a + 2ax + a.n^(n-1)

whats with all the dots

[cerulean saddle]

$a_1 + 2a_2x + 3a_3x^2 + ... + na_nx^{n - 1}$

[midnight finch]

Techie Literate

[cerulean saddle]

Did you... not do series yet?

[solemn mesa]

im sure i did but itwas like half a year ago

i move with the seasons, techie literate

new season, new me

[cerulean saddle]

The three dots represent the in-between terms of the sum.

[solemn mesa]

some would say

ok so

[cerulean saddle]

We're only explicitly showing the beginning and the end, but there's a whole middle.

[solemn mesa]

why does 2x/e^x for x->infinity equal 0

[cerulean saddle]

...try direct substitution.

[solemn mesa]

yk what you remind me of techie literate

my ap econ teacher

you answer questions like him

[cerulean saddle]

So what do you get with direct substitution?

[solemn mesa]

i had to look up a khan academy video to know how to do direct substitution

im still watching it

[cerulean saddle]

...you literally just replace x with infinity.

[solemn mesa]

OK

you couldve just SAID THAT

so, perchance, what am i getting when i replace x with infinity

[cerulean saddle]

I did say that. You just didn't know that's what the words I said meant.

[solemn mesa]

perchance

[cerulean saddle]

You tell me.

[solemn mesa]

youre right

[solemn mesa]

perchance YOU tell ME

perchance

[cerulean saddle]

...what's 2 * infinity?

[solemn mesa]

my ap calc teacher always said BIG

thats what she writes

[cerulean saddle]

It's infinity.

[solemn mesa]

oh

well she always equals it to BIG

thats what she writes on her scratch work

BIG

[cerulean saddle]

Okay, whatever.

She was trying to make a point that obviously went over your head.

[solemn mesa]

me using the wrong right really makes me embarrassed and shameful

[solemn mesa]

no i got the point

it means its a big number

yeah im like that

[cerulean saddle]

No, the point is that infinity isn't a number.

[solemn mesa]

oh

is infinity a concept

[cerulean saddle]

Yes.

[solemn mesa]

well i dont believe in concepts

i believe in FACTS

like NUMBERS

[cerulean saddle]

Infinity is what happens when you count up forever without stopping.

[solemn mesa]

i could so do that

[cerulean saddle]

...facts are also a concept. So are numbers.

[solemn mesa]

oh

um

so

whats e ^ infinity

[cerulean saddle]

You're immortal?

[solemn mesa]

practically

when i dream hard enough

[cerulean saddle]

You tell me. What do you get when you multiply e by itself over and over and over without ever stopping?

[solemn mesa]

like idk what even is e

NO geniunely

what the hell is e

e pisses me off

[cerulean saddle]

...it's approximately 2.718.

[solemn mesa]

oh

so itd be infinity

!!

[cerulean saddle]

Therefore?

[solemn mesa]

e is ummm infinity

wait

but 2x is also infinity

[cerulean saddle]

No.

[solemn mesa]

so shouldnt that

[cerulean saddle]

Right, so?

[solemn mesa]

why does that equal 0 shouldnt it be 1

now im getting sad

[cerulean saddle]

...no.

[solemn mesa]

why does it equal 0

[cerulean saddle]

It's an indeterminate form.

What can we do when we have infinity/infinity?

[solemn mesa]

so YOUD USE L HOSPITAL

[cerulean saddle]

Yes.

[solemn mesa]

you can um cancel it out

[cerulean saddle]

...no. We do this.

[solemn mesa]

ok

but i already did that

with the original function

[cerulean saddle]

...yes. And we can do it again.

[solemn mesa]

so itd be 2 OVER OMHG

OMGOHN

I GET IT

I GET IT

DONT SAY ANYHITNG

I CAN EXPLAIN IT

because 2 over infinity is so

[cerulean saddle]

You can use L'Hopital's Rule as many times in a row as you like as long as the conditions are met.

[solemn mesa]

is practically 0

omg ig et

omg im LITERALLY busting it down rn

god ur so great techie literate

[cerulean saddle]

I mean, there actually is a precise definition of a limit as well, but I get the feeling that maybe you're approaching your limit for today. Or maybe that's me.

[solemn mesa]

um i actually am NOT reaching my limit

im just getting started

i have 10 units of review to do

and im on unit 1

[cerulean saddle]

So then it's me.

[solemn mesa]

<3

[cerulean saddle]

Here's a video about limit definitions. https://youtube.com/watch?v=onrBRLXefuQ

[solemn mesa]

yk what it is techie literate