Limits Question

[strong scarab]

Can someone explain to me why the limit went to the exponent for e^f(x)

[warm plover]

The relation f(limit) = limit(f) for a function f is always true when the function is continuous (and when the limit exists ofc)

Hence if f tends to 2 then exp(f) will tend to exp(f(2))

[strong scarab]

im confused at the f(limit) = limit(f) part

[warm plover]

In other words it's a composition rule for limits

Here you have exp(f(x))

[strong scarab]

e^f(x) yeah

[warm plover]

And you know the limit of the inside, because f(x) -> f(2) when x goes to 2

[strong scarab]

the limit of f(x) is given with the question

[warm plover]

Let's call this value y

y=f(2)

[strong scarab]

it gives the lim x->2 f(x) = 0

[warm plover]

Okay then y=0

Then, you put this in exp

So you want to know the limit of exp(y) when y approaches 0

[strong scarab]

I think what tripping me up was the fact that I did questions such as lim x->5 of x^3 becoming [lim x->5 (x)]^3 (composition rule)

like in this part the limit goes to the inside function

[warm plover]

Yep, this is the same idea

[strong scarab]

yeah the main difference is that the limit went to the exponent

[warm plover]

Oh

But the exponentiation is also a function

Like the cube function

[strong scarab]

oh so technically is should be written as [lim x->5 (x)]^(lim x->5 (3))

[warm plover]

Remember that $e^y = exp(y)$

[dense pasture]

Twenty

[strong scarab]

what does exp(y) mean

[warm plover]

No no, here 3 doesn't change
The function is x^3 so only the variable x changes (so only there gets the "limit" part)

[warm plover]

Do you know the exponential function?

[strong scarab]

ohh, okay I see that

since its e^x

or e^y

it will go to the exponent

[strong scarab]

we weren't introduced to it much

[warm plover]

That's right

Cause this time it's e^x, so the e doesn't change and the x is the variable so this is where the limit appears

[warm plover]

That's okay, as long as you can accept that there's this function e^x

With a certain constant e

e ≈ 2.7182818 something

[strong scarab]

yup, when you explain the part of the limit going to the variable it all made sense

[strong scarab]

ah alr

thanks for the help ❤️