Course:MATH110/Archive/2010-2011/003/Notes/Properties of limits

Limits are the key tool to calculus. They allow us to talk about asymptotes, continuity, derivatives and so much more. Their precise description ( described in this note) involves some fairly technical language, which becomes fairly intuitive after a while, but which makes computing limits quite complicated.

To help out, the community of mathematicians have discovered some very important properties of limits which can often simplify our lives. These properties all have a formal and rigorous proof (involving these $\epsilon$ and $\delta$) that you don't need to know. What you need to know is how to use these properties to ease your computations.

Properties of limits

Let $f$ and $g$ be functions which both have a limit at $x=a$. This means that both

$\displaystyle {\lim _{x\rightarrow a}f(x)}$    and    $\displaystyle {\lim _{x\rightarrow a}g(x)}$    exist.

Let $c$ be any real number.

Then, the following properties are true:

• $\displaystyle {\lim _{x\rightarrow a}\left[f(x)+g(x)\right]=\lim _{x\rightarrow a}f(x)+\lim _{x\rightarrow a}g(x)}$

• $\displaystyle {\lim _{x\rightarrow a}\left[f(x)-g(x)\right]=\lim _{x\rightarrow a}f(x)-\lim _{x\rightarrow a}g(x)}$

• $\displaystyle {\lim _{x\rightarrow a}\left[c\cdot f(x)\right]=c\cdot \lim _{x\rightarrow a}f(x)}$

• $\displaystyle {\lim _{x\rightarrow a}\left[f(x)\cdot g(x)\right]=\lim _{x\rightarrow a}f(x)\cdot \lim _{x\rightarrow a}g(x)}$

• $\displaystyle {\lim _{x\rightarrow a}{\frac {f(x)}{g(x)}}={\frac {\lim _{x\rightarrow a}f(x)}{\lim _{x\rightarrow a}g(x)}}}$    if    $\displaystyle {\lim _{x\rightarrow a}g(x)}\neq 0$