THEOREM Limit of a Quotient If $f$ and $g$ are functions for which $\lim _{x \rightarrow c} f(x)$ and $\lim _{x \rightarrow c} g(x)$ both exist, then $\lim _{x \rightarrow c}\left[\frac{f(x)}{g(x)}\right]$ exists and $\lim _{x \rightarrow c}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim _{x \rightarrow c} f(x)}{\lim _{x \rightarrow c} g(x)}$ provided $\lim _{x \rightarrow c} g(x) \neq 0$ #### Corollary: Limit of a Rational Function If the number $c$ is in the domain of a rational function $R(x)=\frac{p(x)}{q(x)}$, then $\lim _{x \rightarrow c} R(x)=R(c)$