Prove that if $ f \in L^1[0, +\infty]$, then $\lim\limits_{n \to \infty}\frac{1}{n}\int_0^n xf(x)=0$
35325
Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
128882
Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$
53688
How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?
31836
Proof of $\lim_{n\to \infty} \sqrt[n]{n}=1$
64571
Prove that $\int_{E}f =\lim \int_{E}f_{n}$
144273
Find $\lim_{n\to\infty} \frac{1}{n} \int_0^\infty f(x) e^{-a_n x} dx$
81
$\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$
41037
Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions
50682
Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
74286
Prove: $\lim_{n\to\infty} \frac{\sum^n_{k=1} \frac{1}{k}}{ \ln(n) } = 1$
34308
Given $|z|<1$ prove $\lim_{n \to \infty} z^n=0$.
149586
If $\lim_{x\to\infty} f'(x) = L$, prove that $\lim_{x\to\infty} f(x+c)-f(x) = cL$ and $\lim_{x\to\infty} f(x)/x=L$
5728
How to prove $\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n} - 1) = 0$?
117906
Show that $\liminf\limits_{x\to \infty}|f'(x)|=0$ if $\lim\limits_{x\to \infty}f(x)/x=0$ and $f$ is differentiable
74170
Prove that $\lim_{x \to +\infty} \frac{f(x)}{x} = L$ if $\lim_{x \to +\infty} [f(x+1) - f(x)] = L \space$
41515
fine the limit : $\lim_{ n \to \infty }\frac{1}{n}\int_{0}^{n}{ \frac{x\ln(1+\frac{x}{n})}{1+x}}=?$
19986
How can this be proved $\lim_{x\to\infty}(f(x)+f'(x))=l$