L'Hôpital's Rule/Corollary 1

Corollary to L'Hôpital's Rule
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

Suppose that $\forall x \in \openint a b: \map {g^{\prime} } x \ne 0$. Suppose that $\exists c \in \openint a b: \map f c = \map g c = 0$.

Then:
 * $\displaystyle \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \lim_{x \mathop \to c} \frac {\map {f^{\prime} } x} {\map {g^{\prime} } x}$

provided that the second limit exists.

Proof
This follows directly from the definition of limit.

If $\displaystyle \lim_{x \mathop \to c} \frac {\map {f^{\prime} } x} {\map {g^{\prime} } x}$ exists, it follows that:
 * $\displaystyle \lim_{x \mathop \to c} \frac {\map {f^{\prime} } x} {\map {g^{\prime} } x} = \lim_{x \to c^+} \frac {\map {f^{\prime} } x} {\map {g^{\prime} } x}$

That is, if there exists such a limit, it is also a limit from the right.