L’Hospital’s Rule

Definition: L’Hopital’s Rule

If you have two functions, $f\left(x\right) \mathrm{and} g\left(x\right)$, that are differentiable at x = a and the limit of their quotient is an indeterminate form, $\underset{x\to a}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{0}{0}\mathit{ or }\underset{x\to a}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\infty }{\infty },\mathit{ then }\underset{x\to a}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\mathit{lim}}\frac{f\prime \left(x\right)}{g\prime \left(x\right)}$

Everyday Language:

Essentially what L’Hopital’s Rule states is that if you have a limit problem, and the result of that initial limit comes out as one of these two answers, $\underset{x\to a}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{0}{0}\mathit{ or }\underset{x\to a}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\infty }{\infty }$, indeterminate forms, then that initial limit will have the same answer as the new limit of the derivative of the top and bottom equations taken individually , $\underset{x\to a}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\mathit{lim}}\frac{f\prime \left(x\right)}{g\prime \left(x\right)}$

You WILL NOT be doing a quotient rule. You take the derivative of the top equation, $f\left(x\right)$, and the bottom equation, $g\left(x\right)$, separately. The new limit of the derivative equations, $\underset{x\to a}{\mathit{lim}}\frac{f\prime \left(x\right)}{g\prime \left(x\right)}$, will give you the same answer as the limit of the initial limit, $\underset{x\to a}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}$.

NOTE: While the rule is given using a 1 ^st Derivative , you can continue to apply the rule (i.e., keep taking derivatives ) as long as you meet the indeterminate form requirement.