I'm trying to prove that limu→∞umeu=0 for any integer m. I tried using the Ratio Test For Limits to prove that the limit converges to 0 for all real values of m, and therefore it must converge to 0 for all integer values of m since \mathbb{Z} \subset \mathbb{R}. However, using the Ratio Test only seems to complicate things. Is there a better way to show this?
Additionally, I need to prove that \lim_{x\to 0^+}{x\log x}=0 by setting u=-\log x and using the above limit. So far, I have done the following:
\lim_{-\log x\to \infty}{\frac{(-\log x)^m}{e^{-\log x}}}=\lim_{x\to 0^+}x{(-\log x)^m}
but I don't know where to go from here. Also, I cannot use L'Hopital's rule, since I haven't covered it in lectures. Thanks in advance.
Answer
If you accept that e^{u}=\sum_{n=0}^{\infty} \frac{u^n}{n!}, then
\frac{u^m}{e^u}=\frac{u^m}{\sum_{n=0}^{\infty} \frac{u^n}{n!}}\le (m+1)!\frac{u^m}{u^{m+1}}\to 0
as u\to\infty.
No comments:
Post a Comment