The algebra of limits.

Theorem 3.7.1       

$\underset{n \rightarrow \infty}{\text{lim}}u_n$	$\underset{n \rightarrow \infty}{\text{lim}}v_n$	$\underset{n \rightarrow \infty}{\text{lim}}(u_n+v_n)$	$\underset{n \rightarrow \infty}{\text{lim}}(u_n v_n)$	$\underset{n \rightarrow \infty}{\text{lim}} \frac {u_n}{v_n}$	$\underset{n \rightarrow \infty}{\text{lim}}\sqrt{u_n}$
$l_1$	$l_2 \neq 0$	$l_1+l_2$	$l_1l_2$	$\frac {l_1}{l_2}$	$\sqrt{l_1}$ if $l_1>0$
$l_1 \neq 0$	0	$l_1$	0	$\infty$ or no limit	$\sqrt{l_1}$ if $l_1>0$
0	0	0	0	undeterminate	0 if all terms are non negative
$l\neq 0$	$\infty$	$\infty$	$\infty$	0
0	$\infty$	$\infty$	undeterminate	0
$\infty$	$l\neq 0$	$\infty$	$\infty$	$\infty$
$\infty$	0	$\infty$	undeterminate	either $\infty$ or no limit	$+\infty$ if ...
$\pm \infty$	$\pm \infty$	$\pm \infty$	$+\infty$	undeterminate	$+\infty$ if ...
$\pm \infty$	$\mp \infty$	undeterminate	$-\infty$	undeterminate	$+\infty$ if ...

In particular, the set of convergent sequences (with the same set ofindices) is a real vector space.

For limits of functions, we have a similar table; v.i. Theorem 4.1.