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 $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Suppose that $\forall x \in \left({a \,.\,.\, b}\right): g^{\prime} \left({x}\right) \ne 0$. Suppose that $\exists c \in \left({a \,.\,.\, b}\right): f \left({c}\right) = g \left({c}\right) = 0$.

Then:
 * $\displaystyle \lim_{x \to c} \frac {f \left({x}\right)} {g \left({x}\right)} = \lim_{x \to c} \frac {f^{\prime} \left({x}\right)} {g^{\prime} \left({x}\right)}$

provided that the second limit exists.

Proof
This follows directly from the definition of limit.

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

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

However, this result was in fact discovered by.

Because of variants in the rendition of his name, this result is often seen written as L'Hospital's Rule.