Continuous completion of an example function quotient ${\displaystyle {\frac {f(x)}{g(x)}}}$ by l'Hôpital's rule, for example functions ${\displaystyle f(x)=sin(x)}$ and ${\displaystyle g(x)=-0.5x}$:

At ${\displaystyle x=0}$, the quotient is undefined since ${\displaystyle g(0)=0}$. Defining the function as ${\displaystyle \left\{{\begin{array}{ll}{\frac {f(x)}{g(x)}}&{\mbox{ if }}x\neq 0\\{\frac {f'(x)}{g'(x)}}&{\mbox{ if }}x=0\\\end{array}}\right.}$ completes the function definition such that a continuous function results, by l'Hôpital's rule.