Question

Show that the two functions grow at the same rate:
sqrt(9^x + 2^x) and 3^x

Answer 1

I've tried a ton to manipulate the limit, but I just can't get it to turn into a constant that isn't 0

Answer 2

I tried squaring the part inside of the limit, but that didn't help all that much

Answer 3

oh nvm

Answer 4

This may or may not use l'hopital btw

Answer 5

yeah you can use l'hopital's rule to find that the\[\lim_{x \rightarrow \infty}(\sqrt{9^x+2^x)}/(3^x)) = 1\]
If you just plug in infinity for x you'll get infinity/infinity which is undefined, so you have to use l'hopital's rule ( differentiate the top and bottom until you dont get infinity/infinity)

Answer 6

Except that by using l'Hopital, it overcomplicates the problem a tremendous amount

Answer 7

You get all of these nasty natural logs in there, in addition to the regular exponential functions that are already in the problem, so it only makes the problem worse

Answer 8

@e.mccormick If you can help me that would be greatly appreciated

Answer 9

No matter what I did, I just could not get that to simplify inside of the limit :/

Answer 10

you don't need to use L'Hospitals rule

Answer 11

Yeah, I kinda figured because it wasn't going to help at all

Answer 12

put the \(3^x\) under the square root

Answer 13

Ok, so that would make it 3^2x right?

Answer 14

you can write it another way

Answer 15

How?

Answer 16

functions grow at the same rate if
\[\lim_{x \rightarrow \infty} \frac{ f(x) }{g(x) } = M \neq 0\] where M is a finite non zero number

Answer 17

Yes

Answer 18

I need the limit to check out to a non-zero constant

Answer 19

I cheated but there you go

Answer 20

There was some black magic in that solution

Answer 21

you can also write it as

\[\frac{\sqrt{9^x+2^x}}{3^x}=\sqrt{\frac{9^x+2^x}{9^x}}=\sqrt{1+\left(\frac{2}{9}\right)^x}\]

Answer 22

where was the black magic lmao

Answer 23

Whooooa

Answer 24

Oh my god

Answer 25

Can you just pull down the 2 from the exponent like that to square the 3?

Answer 26

I feel like wafflemans brain just got blown

Answer 27

Because that is some crazy stuff right there

Answer 28

I just became a toasted waffle

Answer 29

\[(3^x)^2=3^{2x}=(3^2)^x=9^x\]

Answer 30

Mind blown level=Hiroshima

Answer 31

basically he is saying
\[\sqrt{9^{x}} = 3^{x}\]

Answer 32

yeah what zarkon said lol

Answer 33

That is crazy. Thanks guys. I never would have even considered doing that.

Answer 34

I tried to simplify myself and gave up because I never would have gotta that stuff either tbh
sucks to be you

Answer 35

Hahaha. BC calc IS pretty miserable at times