Infinitely Large Sequences and Infinite Limits

$x_{n}=(-1)^{n} n$;

$x_{n}=2^{\sqrt{n}}$;

$x_{n}=\lg (\lg n) \quad (n \geq 2)$

$\lim \limits_{n \to \infty } x_{n}=\infty$;

$\lim \limits_{n \to \infty } x_{n}=-\infty$;

$\lim \limits_{n \to \infty } x_{n}=+\infty$.

$\lim_{n \rightarrow \infty } n = +\infty$;

$\lim_{n \rightarrow \infty }(5-0,5 n) = -\infty$;

$\lim_{n \rightarrow \infty }(-1)^{n} n = \infty$;

$\lim_{n \rightarrow \infty } a^{n} = \infty$, where $\left|a\right| > 1$;

$\lim_{n \rightarrow \infty } \sqrt[3]{n-100} = +\infty$;

$\lim_{n \rightarrow \infty } n^{1 / p} = +\infty$, where $p \in \mathbb {N}$;

$\lim_{n \rightarrow \infty } \lg n = +\infty$;

$\lim_{n \rightarrow \infty } \log_{a} n = +\infty$, where $a > 1$;

$\lim_{n \rightarrow \infty } \log_{a} n = -\infty$, where $0 < a < 1$.

$\lim_{n \rightarrow \infty } n^{2} = +\infty$;

$\lim_{n \rightarrow \infty } n^{p} = +\infty$, where $p \geqslant 1$;

$\lim_{n \rightarrow \infty }(2 n-5)^{5} = +\infty$;

$\lim_{n \rightarrow \infty }(\lg n)^{3} = +\infty$;

$\lim_{n \rightarrow \infty }\left(\log_{a} n\right)^{p} = +\infty$, where $a > 1, p \geqslant 1$;

$\lim_{n \rightarrow \infty }\left(0,5-(-1)^{n} \sqrt[3]{n}\right) = \infty$;

$\lim_{n \rightarrow \infty }(-n)^{n} = \infty$;

$\lim_{n \rightarrow \infty }(4 \sqrt{n}-n) = -\infty$.

Let $x_{n} \neq 0, \lim_{n \rightarrow \infty } x_{n} = 0$. Prove that $\lim_{n \rightarrow \infty } \frac{1}{x_{n}} = \infty$.

Let $x_{n} \neq 0, \lim_{n \rightarrow \infty } x_{n} = \infty$. Prove that $\lim_{n \rightarrow \infty } \frac{1}{x_{n}} = 0$.

$\left(\frac{n}{10}\right)^{n}$

$\frac{n^{2}}{n+8}$

$\frac{n !}{4^{n}}$

$\frac{1}{\sqrt[n]{2}-1}$

$\frac{2}{1-\sqrt[n]{n}}$

$\frac{5^{n}}{n^{2}}$

$\frac{a^{n}}{n^{k}}$, where $\left|a\right| > 1, k \in \mathbb {N}$.

Let $\lim_{n \rightarrow \infty } x_{n} = +\infty$ and $y_{n} \geqslant c$ for all $n \in \mathbb {N}$. Prove that $\lim_{n \rightarrow \infty }\left(x_{n}+y_{n}\right) = +\infty$.

Let $\lim_{n \rightarrow \infty } x_{n} = -\infty$ and $y_{n} \leqslant c$ for all $n \in \mathbb {N}$. Prove that $\lim_{n \rightarrow \infty }\left(x_{n}+y_{n}\right) = -\infty$. that:

Let $\lim_{n \rightarrow \infty } x_{n} = a$, where $a$ is $+\infty$ or $-\infty$. Prove that:

1. if for all $n$ from some index onward, $y_{n} \geqslant c > 0$, then $\lim_{n \rightarrow \infty } x_{n} y_{n} = a$;

2. if for all $n$ from some index onward, $y_{n} \leqslant c < 0$, then $\lim_{n \rightarrow \infty } x_{n} y_{n} = -a$.

Let $\lim_{n \rightarrow \infty } x_{n} = \infty$ and for all $n$ from some index onward, $\left|y_{n}\right| \geqslant c > 0$. Prove that $\lim_{n \rightarrow \infty } x_{n} y_{n} = \infty$.

$\lim_{n \rightarrow \infty }(5 / n-n) = -\infty$;

$\lim_{n \rightarrow \infty }(\lg n+2 \cos \pi n) = +\infty$;

$\lim_{n \rightarrow \infty } \frac{n \sqrt{n}}{n+1} = +\infty$; $b \in \mathbb {R}$;

$\lim_{n \rightarrow \infty }(a n+b) = +\infty$ for $a > 0, \quad \lim_{n \rightarrow \infty }(a n+b) = -\infty$ for $a < 0$,

$\lim_{n \rightarrow \infty } \frac{n^{4}-100 n}{n^{2}+100} = +\infty$;

$\lim_{n \rightarrow \infty }\left(\sqrt{n^{2}-1}-\sqrt{n}\right) = +\infty$;

$\lim_{n \rightarrow \infty } \frac{1}{\sqrt{n\left(\sqrt{n^{2}-1}-n\right)}} = -\infty$.

$\lim_{n \rightarrow \infty } x_{n} = \infty$;

$\lim_{n \rightarrow \infty } x_{n} = +\infty$;

$\lim_{n \rightarrow \infty } x_{n} = -\infty$

$\lim_{n \rightarrow \infty } x_{n} = \infty$;

$\lim_{n \rightarrow \infty } x_{n} = +\infty$;

$\lim_{n \rightarrow \infty } x_{n} = -\infty$.

every infinitely large sequence is unbounded;

every unbounded sequence is infinitely large?

the sequence does not tend to $\infty$;

the sequence does not tend to $+\infty$;

the sequence does not tend to $\mathbf{\kappa }-\infty$.

$x_{n} = n^{2} \cos (\pi n / 2)$;

$x_{n} = n^{(-1)^{n}}$

$x_{n} = \frac{n}{1+n \sin (\pi n / 2)}, n \in \mathbb {N}$.

$\lim_{n \rightarrow \infty } \frac{n^{3}-n+3}{n^{2}+1} = +\infty$;

$\lim_{n \rightarrow \infty } \frac{n^{3}-100}{n^{2}+100} = +\infty$.

$+\infty$;

$-\infty$;

a finite limit?

a finite limit;

an infinite limit?

$\lim_{n \rightarrow \infty } \sqrt{n !} = +\infty$;

$\lim_{n \rightarrow \infty } 2^{\sqrt{n-1}} = +\infty$;

$\lim_{n \rightarrow \infty } \frac{n^{2}}{\log_{a}(n+1)} = +\infty , a > 1$;

$\lim_{n \rightarrow \infty } \frac{\ln n !}{n} = +\infty$.

$\lim_{n \rightarrow \infty } \frac{n}{\sqrt{n}-\sqrt{n+1}} = -\infty$;

$\lim_{n \rightarrow \infty } \sqrt{n^{3}}(\sqrt{n+1}-\sqrt{n}-\sqrt{n-1}) = -\infty$.

$+\infty$;

a finite limit? In the second case, find this limit.

$\frac{(n+1)(n+2) \ldots (2 n-1) 2 n}{n^{n}}$;

$\sqrt[n]{n !}$;

$\left(1+\frac{1}{n}\right)^{n^{2}}$;

$\left(1+\frac{1}{2^{n}}\right)^{n !}$;

$\frac{1}{n}(1+\sqrt{2}+\ldots +\sqrt{n})$.

$\lim_{n \rightarrow \infty } x_{n} y_{n} = \infty$ (respectively equals $+\infty ,-\infty$) for $x > 0$;

$\lim_{n \rightarrow \infty } x_{n} y_{n} = \infty$ (respectively equals $-\infty ,+\infty$) for $x < 0$.

if $\left|y_{n}\right| \leqslant C, n \in \mathbb {N}$, then $\lim_{n \rightarrow \infty } x_{n} y_{n} = \infty$;

if $y_{n} \geqslant x_{n}, n \in \mathbb {N}$, then $\lim_{n \rightarrow \infty } y_{n} = \infty$;

if $\lim_{n \rightarrow \infty } y_{n} = \infty$, then $\lim_{n \rightarrow \infty }\left(x_{n}+y_{n}\right) = \infty$?

Let

or

Prove that $\lim_{n \rightarrow \infty } x_{n} y_{n} = +\infty$.

Let

Prove that $\lim_{n \rightarrow \infty } x_{n} y_{n} = -\infty$.

$\lim_{n \rightarrow \infty }\left(x_{n}-y_{n}\right) = +\infty$;

$\lim_{n \rightarrow \infty }\left(x_{n}-y_{n}\right) = 1$;

$\lim_{n \rightarrow \infty }\left(x_{n}-y_{n}\right) = -\infty$;

the sequence $\left\{ x_{n}-y_{n}\right\}$ has neither a finite nor an infinite limit.

$\lim_{n \rightarrow \infty } \frac{x_{n}}{y_{n}} = 0$

$\lim_{n \rightarrow \infty } \frac{x_{n}}{y_{n}} = 1$;

$\lim_{n \rightarrow \infty } \frac{x_{n}}{y_{n}} = +\infty$;

the sequence $\left\{ x_{n}-y_{n}\right\}$ has neither a finite nor an infinite limit.

$\lim_{n \rightarrow \infty } x_{n} y_{n} = 0$;

$\lim_{n \rightarrow \infty } x_{n} y_{n} = 1$;

$\lim_{n \rightarrow \infty } x_{n} y_{n} = \infty$;

the sequence $\left\{ x_{n} y_{n}\right\}$ has neither a finite nor an infinite limit.

Prove that if the sequences $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ are such that

then

Prove that the converse statement also holds.

Let $\lim_{n \rightarrow \infty } x_{n} = +\infty , m_{n} = \inf_{k \geqslant n}\left\{ x_{k}\right\} , n \in \mathbb {N}$. Prove that $\lim_{n \rightarrow \infty } m_{n} = +\infty$.

Let $\lim_{n \rightarrow \infty } x_{n} = -\infty , M_{n} = \sup_{k \geqslant n}\left\{ x_{k}\right\} , n \in \mathbb {N}$. Prove that $\lim_{n \rightarrow \infty } M_{n} = -\infty$.