# L'Hôpital's Rule For any two differentiable functions $f$ and $g$, if $f(a) = g(a) = 0$ or $\pm \infty$, then $\lim_{x \to a} \frac {f(x)}{g(x)} = \lim_{x \to a} \frac {f'(x)}{g'(x)} = \frac {f'(a)}{g'(a)}$ ^853loo ## Reasoning As $x$ gets closer and closer to $a$, the ratio of the two functions approximates the ratio of their linear approximations (first two elements of their [[Taylor Series]]): $F(x, a) = \lim_{x \to a} \frac {f(x)}{g(x)} = \frac {f(a)+f'(a)(x-a)}{g(a)+g'(a)(x-a)}=\frac {f'(a)(x-a)}{g'(a)(x-a)}=\frac {f'(a)}{g'(a)}$ ($f(a) = g(a) = 0$ or $\pm \infty$, per requirements)