# Proof of special case of l'Hôpital's rule

L'Hôpital's rule helps us find limits in the form $\displaystyle\lim_{x\to c}\dfrac{u(x)}{v(x)}$ where direct substitution ends in the indeterminate forms $\dfrac00$ or $\dfrac\infty\infty$.
The rule essentially says that if the limit $\displaystyle\lim_{x\to c}\dfrac{u'(x)}{v'(x)}$ exists, then the two limits are equal:
$\displaystyle\lim_{x\to c}\dfrac{u(x)}{v(x)}=\displaystyle\lim_{x\to c}\dfrac{u'(x)}{v'(x)}$
The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.