Limits

[latent cosmos]

Instructor never covered this in class... need help

[lapis magnet]

Do you know what the graph of 4^x looks like?

[latent cosmos]

yes

like a backwards L

kinda

[lapis magnet]

Alright.
So, what happens when x is positive and large (so, for x -> +∞)?

[latent cosmos]

ohhh so what happens to y?

it goes to infinity right

[lapis magnet]

Yes.

[latent cosmos]

oh oh oh ok

i get it now

[lapis magnet]

And what about large negative numbers (x -> -∞)?

[latent cosmos]

it goes to 0

[lapis magnet]

Yes.

[latent cosmos]

well the limit

[lapis magnet]

So, the correct option is the selected one.

[latent cosmos]

one more question

[lapis magnet]

Yes?

[latent cosmos]

instructor never taught us x -> to infinity

just x to whole numbers

[lapis magnet]

So, here is how you deal with limits of rational functions when x -> +∞ or x -> -∞.
First of all, determine the power of the polynomial in the numerator and denominator.

[latent cosmos]

2

[lapis magnet]

Yes, both 2.
Then, you need to divide each term in the numerator and the denominator by x to the maximum power. In this case, by x^2. What will you get?

[vivid swan]

then you use the identity that n/x x→∞ is 0

[lapis magnet]

Yes.

[vivid swan]

leaving 7/14 aka ½

[lapis magnet]

Basically, all the terms except the leading ones will have the form const/x^n, so their limit will be 0.
Thus, only the leading terms determing the value of the limit.

[latent cosmos]

i got (7 - 6/x - 17/x^2) / (14 + 2/x - 11/x^2)

[vivid swan]

now you do direct sub

[lapis magnet]

Yes, that's correct.
So, now observe that everything except 7 and 14 goes to 0 when x -> ∞.

[vivid swan]

6/∞ is 0 and 17/∞² is 0

so on

[latent cosmos]

so the answer is 1/2?

[lapis magnet]

Yes.

[latent cosmos]

can you use this method everytime there is a polynomial?

[vivid swan]

yes

[latent cosmos]

ok bet

thanks