L’Hospital’s_Rule

[!warning] AI-Generated This file was generated by AI and may require review.

Definition

Assume $f$ and $g$ are differentiable on $(a,b)$ and that $g'(x) \neq 0$ for all $x \in (a,b)$. If

$$ \lim_{x \to a^+} f(x) = 0 \quad\text{and}\quad \lim_{x \to a^+} g(x) = 0 $$

then

$$ \lim_{x \to a^+} \frac{f(x)}{g(x)} = \lim_{x \to a^+} \frac{f'(x)}{g'(x)} $$

provided the limit on the right-hand side exists.

The rule also applies to the indeterminate form $\frac{\infty}{\infty}$.

Proof

By the Generalized_Mean_Value_Theorem, for each $x \in (a,b)$ there exists $c_x$ between $a$ and $x$ such that

$$ \frac{f(x)}{g(x)} = \frac{f(x) - f(a)}{g(x) - g(a)} = \frac{f'(c_x)}{g'(c_x)} $$

(using $f(a) = g(a) = 0$ by continuity).

As $x \to a^+$, we have $c_x \to a^+$, so the result follows.