Solving a Long Limit Problem: Finding the Limit of e^x/x^n

[caelestis]

Hello,

I've been given a long limit problem to solve and i got stuck on this part in the question. Could someone please give me hints or suggestions on where to go next?

Given that \frac{e^{x}}{x^{n}} > e^{x - n\sqrt{x}}

Find the lim_{x\rightarrow+\infty} \frac{e^{x}}{x^{n}}



Well i know that lim_{x\rightarrow\infty} e^{x} = \infty

but i think the question would like us to use the inequality above.

Any help would be greatly appreciated :)

 

[quantumdude]

I don't see how the inequality helps. Do you know about power series? If so, then can you write down a power series for the function e^x/x^n, and take its limit?

 

[caelestis]

I don't see how the inequality helps. Do you know about power series? If so, then can you write down a power series for the function e^x/x^n, and take its limit?

umm... i haven't learned power series... Is there another approach i could take??
i tried l'hospitals rule but my answer didn't help much in this...

 

[tiny-tim]

Hi caelestis!

Hint: prove that lim (x - n√x) = ∞.