Using L’Hôpital’s rule for finding limits of indeterminate forms

Proof of special case of l'Hôpital's rule

L'Hôpital's rule helps us find limits in the form lim⁡x→cu(x)v(x)\displaystyle\lim_{x\to c}\dfrac{u(x)}{v(x)} where direct substitution ends in the indeterminate forms 00\dfrac00 or ∞∞\dfrac\infty\infty.
The rule essentially says that if the limit lim⁡x→cu′(x)v′(x)\displaystyle\lim_{x\to c}\dfrac{u'(x)}{v'(x)} exists, then the two limits are equal:
lim⁡x→cu(x)v(x)=lim⁡x→cu′(x)v′(x)\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.
Loading