The algebra of limits.

Theorem 3.7.1       

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

Noah Dana-Picard 2007-12-28