Thursday, 19 October 2017

real analysis - Prove that limutoinftyfracumeu=0




I'm trying to prove that limuumeu=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

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...