Sunday, 25 September 2016

calculus - Limit limxtoinftyxe/ex



I had this problem on my math test, and was stuck on it for quite some time.



lim



I knew that the bottom grew faster than the top, but I didn't know how to prove it. I wrote that the limit approaches 0, but I am not sure how to prove it mathematically.


Answer




Show first that it is in indeterminate form.



Then perform L'Hopital's rule, differentiating the top and bottom.



\lim_{x \to \infty} \frac{x^e}{e^x}= \lim_{x \to \infty} \frac{ex^{e-1}}{e^x}=e(e-1)(e-2)\lim_{x \to \infty} \frac{1}{x^{3-e}e^x}=0


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