L' Hospital's Rule

analysis

Definition

L' Hospital's Rule

Let the functions $f$ and $g$ be continuous on $[a,b]$ and differentiable in the interior $(a,b)$. Furthermore, let $x_0\in [a,b]$ and let $f(x_0)=g(x_0)=0$. Also, it is assumed that the limit ${\lim }_{x\to x_0}\frac{f^{\prime }(x)}{g^{\prime }(x)}$ exists (or the one-sided limit, if $x_0=a$ or $x_0=b$). Then the following holds:

$$\underset{x\to x_0}{\lim }\frac{f(x)}{g(x)}=\underset{x\to x_0}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$$