Question

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.

Answer 1

have you tried rewriting without negative exponent ?

Answer 2

\[\lim_{x \rightarrow \infty}x^9e ^{-x^8}\]= \[\frac{ x^9 }{ e ^{x^8} }\]
=\[\lim_{x \rightarrow \infty}\frac{ 9x^7 }{ 8e ^{x^8} }\]

Answer 3

thats what I have so far

Answer 4

that should make it obvious what the limit is...
but if not I guess you could l'hospital what about (9+1) times and if that 9 l'hospital version of the limit doesn't convince you that the limit is ____ then I don't know what will :p

Answer 5

eventually you will get a constant on top and you will have an exponential still on bottom

Answer 6

so would I have to find the derivative around 7-8 more times until its just a constant at top?

Answer 7

\[(e^{x^8})'=(x^8)' \cdot e^{x^8}=8x^7 \cdot e^{x^8}\]
I lied you won't need that many derivatives

Answer 8

but you will still wind up with a constant over a function going to infinity still after [,e .[re derivatove

Answer 9

after one more derivative *

Answer 10

@Zenmo you should have 9x^8 up top after using L' Hospitals rule

and `-8x^7*e^(-x^8)` in the bottom. Then simplify

Answer 11

negative won't be there though

Answer 12

because we put in bottom

Answer 13

sorry, `8x^7e^(x^8)` in the bottom

Answer 14

yes true

Answer 15

Just want to say that you could work it using Lhopital rule just one time

Answer 16

but ofcourse it requires a clever but elementary trick...

Answer 17

[,e .[re derivatove =one more derivative lol
my keyboard was skewed to the right when I was typing

Answer 18

Here is a neat way oldrin had shown @ganeshie8 can recall to

\(\color{blue}{\text{Originally Posted by}}\) @oldrin.bataku
\[\frac{x^n}{e^x}=n^n\left(\frac{x/n}{e^{x/n}}\right)^n\] so consider passing the limit in: \[\lim_{x\to\infty} n^n\left(\frac{x/n}{e^{x/n}}\right)^n=n^n\left(\lim_{u\to\infty}\frac{u}{e^u}\right)^n\] with \(u=x/n\)
\(\color{blue}{\text{End of Quote}}\)

Answer 19

lol you have an elephant memory

Answer 20

Hehe =P