Question

I still have a question on a part of this:
Show that the two functions grow at the same rate:
sqrt(9^x + 2^x) and 3^x

Answer 1

@amorfide

Answer 2

what question

Answer 3

Ok so when you get to \[\sqrt{\lim_{x \rightarrow \infty} 1+(2/9)^{x}}\] how do you prove that that goes to a constant?

Answer 4

well you know the limit of 2/9 to the power of x goes to 0 because a decimal to the power of x will tend to 0 as x tends to infinite
so you just get the limit of root 1 as x tends to infinite so you just get 1 as the limite
therefore the limit of f(x)/g(x) as x tends to infinite is 1 meaning that the growth rate is the same

I think that was clear idk

Answer 5

I understand, but is there a mathematical way to prove it?

Answer 6

I mean like to algebraically simplify it

Answer 7

what do you mean
like
an actual proof?

Answer 8

Well to simplify the (2/9)^x to 0

Answer 9

Or do I not have to?

Answer 10

http://prntscr.com/6ggqj2

Answer 11

there is the working out as to why

Answer 12

How did you go to the part that is above the gray part to the gray part? (if you know what I mean at all)

Answer 13

How did you do these 3 lines http://prntscr.com/6ggsfr

Answer 14

idk how to explain the continuity thing uhm @e.mccormick @Nnesha

Answer 15

Its alright :/

Answer 16

@texaschic101

Answer 17

@jim_thompson5910

Answer 18

if f(x) is continuous around x=g(a) and g(x) is continuous around x=a, then
\[\lim_{x \rightarrow a} f(g(x)=f(\lim_{x \rightarrow a}g(x))\]

compute \[e ^{\lim_{x \rightarrow infty}(xln(2/9))}\]

\[\lim_{x \rightarrow \infty}(xln (\frac{ 2 }{ 9 }))=-\infty \]

Answer 19

sorry the coding messed up on that limit with e

Answer 20

Should the e be after the limit?

Answer 21

no the limit is in the power

Answer 22

if it is the negative infinite that you are confused about then

Answer 23

Yes

Answer 24

oops some of it was cut off

Answer 25

Is it because ln(2/9) is negative?

Answer 26

http://prntscr.com/6ggygk

Answer 27

yeah

Answer 28

because it would be like -ln(9/2) or something

Answer 29

so you get -infinite

Answer 30

Ok. And in the original thing that you posted here, you can raise the thing to the e power and take the ln of the stuff because they cancel out right?

Answer 31

http://prntscr.com/6ggqj2

Answer 32

yeah because e and the ln cancel that means it still turns out to be (2/9)^x
but written in another form that is easier to get a limit from

Answer 33

Ok. I get it now! Thank you so much!!!

Answer 34

sorry for not being elaborate in the first place lol

Answer 35

No worries. I don't think that my teacher will make us explain that far, but its hard to tell in this chapter what we have to prove and what we can assume that they know

Answer 36

I think it is safe to say that aslong as you simplified it to something you can get the limit of
such as the e^x format
you can just say what the limit is, because you showed the understanding, but you are right, better safe than sorry

Answer 37

Ok. Thanks again!