L'Hôpital's Rule/Corollary 1

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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.

$\blacksquare$


Source of Name

This entry was named for Guillaume de l'Hôpital.


However, this result was in fact discovered by Johann Bernoulli.

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