Tell what theorems are available for calculating limits of sequences?

There are some theorems for calculating limits of sequence.
Let ${\displaystyle a_n}$ and ${\displaystyle b_n}$ be sequence of real numbers, and let A and B be real numbers. These rules hold if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=A}$ and ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{b}_n=B}$
Sum rule: ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(a_n+b_n)=A+B}$
Difference rule: ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(a_n-b_n)=A-B}$
Constant multiple rule: ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(k\times b_n)=k\times B}$ (any number k)
Product rule: ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(a_n\times b_n)=A\times B}$
Quotient rule: ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=\frac{A}{B}}$ if ${\displaystyle B\ne 0}$
I can give you some examples:
Constant Rule
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(-\frac{1}{n})=-1\times \underset{n\to \mathrm{\infty }}{lim}\left(\frac{1}{n}\right)}$
${\displaystyle -1\times 0}$
${\displaystyle =0}$
Difference rule:
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\left(\frac{n-1}{n}\right)=\underset{n\to \mathrm{\infty }}{lim}(1-\frac{1}{n})}$
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}1-\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}}$
${\displaystyle =1-0}$
${\displaystyle =0}$
Product Rule:
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\left(\frac{5}{n^2}\right)=5\times \underset{n\to \mathrm{\infty }}{lim}\left(\frac{1}{n}\right)\times \underset{n\to \mathrm{\infty }}{lim}\left(\frac{1}{n}\right)}$
${\displaystyle =5\times 0\times 0}$
${\displaystyle =0}$
Sum and Quotient Rule:
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\left(\frac{4-7n^6}{n^6+3}\right)}$
${\displaystyle =\underset{n\to \mathrm{\infty }}{lim}\left(\frac{\frac{4}{n^6}-7}{1+\frac{3}{n^6}}\right)}$
${\displaystyle =\frac{0-7}{1+0}}$
${\displaystyle -7}$

Theorem 2 is the sandwich Theorem for sequence
Let ${\displaystyle a_n}$, ${\displaystyle b_n}$ and ${\displaystyle c_n}$ be sequences of real numbers. If ${\displaystyle a_n\le b_n\le c_n}$ holds for all n beyond some index N, and if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=\underset{n\to \mathrm{\infty }}{lim}{c}_n=L}$ then ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{b}_n=L}$ also.
Consider the n-th formula for the sequence function is ${\displaystyle a_n=\frac{\sin n}{n}}$
The sine function lies between ${\displaystyle -1}$ and ${\displaystyle 1}$
${\displaystyle -1\le \sin n\le 1}$
Divided by n
${\displaystyle \frac{-1}{n}\le \frac{\sin n}{n}\le \frac{1}{n}}$
If ${\displaystyle n\to \mathrm{\infty }}$ then ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\to 0}$ and ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{-1}{n}\to 0}$
Apply the sandwich theorem:
If ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{-1}{n}=0}$ then the function converges
Thus, the sequence is converge
The limit of the sequence is 0.
Theorem 3. The continuous function theorem for sequences.
Let ${\displaystyle a_n}$ be a sequence of real numbers. If ${\displaystyle a_n\to L}$ and if ${\displaystyle f}$ is a functon that is continuous at ${\displaystyle L}$ and defined at all ${\displaystyle a_n}$, then ${\displaystyle f\left(a_n\right)\to f\left(L\right)}$
For example,
Consider the sequence
${\displaystyle \sqrt{\frac{n+1}{n}}}$
The value of the sequence is 1
The value of ${\displaystyle \frac{n+1}{n}\to 1}$
Consider ${\displaystyle f\left(x\right)=\sqrt{x}}$ and ${\displaystyle L=1}$
${\displaystyle \sqrt{\frac{n+1}{n}}=\sqrt{1}=1}$