$\lim \inf \frac{x_{n+1}}{x_n} \leq \lim \inf \sqrt[n]{x_n}$ for strictly positive real seq $\{x_n\}$
117640
Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
128551
Let a be a positive number. Then $\lim_{n \to \infty}[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}]$
118023
Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of prime numbers, then $\lim_{n \to \infty}\frac{p_{n+1}}{p_n} = 1$?
150995
Show that $\liminf(\sqrt{x_n})=\sqrt{\liminf x_n}$
10902
Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $
5728
How to prove $\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n} - 1) = 0$?
113336
Limit of $(\frac{1}{n^2})^{(\frac{1}{n})}$
144403
Limit $\lim_{x\rightarrow +\infty}\sqrt{x}e^{-x}\left(\sum_{k=1}^{\infty}\frac{x^{k}}{k!\sqrt{k}}\right)$
59584
A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy
97747
Limit of $\lim\limits_{n\to\infty} (1 + \frac{x_n}{n})^n$
15982
Limit of $b_n$ when $b_n=\frac{a_n}{n}$ and $\lim\limits_{n\to\infty}(a_{n+1}-a_n)=l$
49497
$\lim a_n = 0$ if $\left|\frac{a_{n +1}}{a_n}\right|\to L < 1$
64501
Homework: limit of $\frac{1}{n}\sqrt[n]{\prod_{k=1}^n(n+k)}$ as $n\to\infty$
133191
Finding limit using inequalities: $\liminf \frac{a_{n+1}}{a_n} \le \liminf (a_n)^ {1/n}\le\limsup (a_n)^ {1/n}\le \limsup \frac{a_{n+1}}{a_n}$
104996
Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$
60831
Need hint for $\lim_{x\to 0} \frac{(x+1)^\frac{1}{x}-e}{x}$
88782
Finding the limit of $x_1=0 , x_{n+1}=\frac{1}{1+x_n}$