If $x_n\to x$, with $x_n=\frac{p_n}{q_n}$, and $x_n\ne x,n\in\Bbb{N}$, then $\lim_{n\to\infty} q_n=\infty$.
57233
Proof of continuity of Thomae Function at irrationals.
149176
True/False: $\mathop {\lim }\limits_{n \to \infty } {{{a_n}} \over {{b_n}}} = 1$ implies $\sum {{a_n},\sum {{b_n}} } $ converge or diverge together.
66441
Prove that $|x|<1$ implies that $\lim_{n \to \infty} x^n=0$
146528
Define $I_n=\int_0^1\frac{x^n}{\sqrt{x^2+1}}dx$ for every $n\in\mathbb{N}$. Prove that $\lim_{n\to\infty}nI_n=\frac{1}{\sqrt 2}$.
9694
If $(y_{2n}-y_n) \to 0$ then $\lim_{n\to \infty} y_n$ exists
135700
Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
101027
What is the value of lim$_{n\to \infty} a_n$ if $\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}$?
115411
if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$
118023
Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of prime numbers, then $\lim_{n \to \infty}\frac{p_{n+1}}{p_n} = 1$?
155333
Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
101716
Find this limit: $ \lim_{n \to \infty}{(e^{\frac{1}{n}} - \frac{2}{n})^n}$
91523
Prove this limit as $x\to\infty$
109489
How to solve $ \lim_{n\to\infty} \frac {(n!)^\frac{1}{n}}{n} $?
142778
Finding $\lim_{n\to\infty}a_n$ if $a_0=a,a_1=b$ and $a_n=\frac{1}{2}(a_{n-1}+a_{n-2})$ where $a, b \in \mathbb{R}$
11961
Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$?
131425
Prove that the convergence of $\prod_{n=1}^{\infty} b_n$ implies $\lim_{n→∞} b_n = 1$
9781
How to prove that $\lim_{n \to \infty} n x^{n} = 0 $ when $0<x<1$?