Hospital's Rule: Understanding and Application

The Hospital formula states that if \( \lim\limits_{x\to a} f(x)=\lim\limits_{x\to a}g(x)=0 \) or \( \pm \infty \), but

$$ \lim\limits_{x\to a} \frac{f'(x)}{g'(x)} \qquad \text{exists.} $$

Then

$$\lim\limits_{x\to a} \frac{f(x)}{g(x)}=\lim\limits_{x\to a} \frac{f'(x)}{g'(x)}.$$

In some books also written as: If \( h(x)=\frac{f(x)}{g(x)}\), \(\lim\limits_{x\to a} f(x) = \lim\limits_{x\to a} g(x) = 0\), \( g'(x) \ne 0 \), and one-sided derivatives of a quotient \( [h'(x^+), h'(x^-)]\) or \( h'_-(x)=h'_+(x)=L \), then $$ \lim\limits_{x\to a} \frac{f(x)}{g(x)}= \lim\limits_{x\to a} h(x)=\lim\limits_{x\to a} \frac{f'(x)}{g'(x)}=L.$$