TheoremProved

L'Hôpital's rule

Statement

Suppose $f,g$ are differentiable near $a$ (except possibly at $a$ ), $g'(x)\neq 0$ nearby, and either $\lim_{x\to a} f(x)=\lim_{x\to a}g(x)=0$ or both tend to $\pm\infty$ . If $\lim_{x\to a}\frac{f'(x)}{g'(x)}$ exists (or is $\pm\infty$ ), then $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$ .

Why is it true?

Near a tied '0/0' race between two quantities both shrinking to nothing, it's not the sizes but the relative speeds at which they vanish that decide the limit of their ratio - whichever one is shrinking 'faster' (as measured by derivatives) dominates.

Proof sketch

Extend $f,g$ by setting $f(a)=g(a)=0$ (in the $0/0$ case), so both are continuous at $a$ . For $x$ near $a$ , apply the Cauchy mean value theorem to $f,g$ on the interval between $a$ and $x$ : $\frac{f(x)-f(a)}{g(x)-g(a)}=\frac{f'(\xi)}{g'(\xi)}$ for some $\xi$ strictly between $a$ and $x$ . As $x\to a$ , $\xi\to a$ too, so $\frac{f(x)}{g(x)}=\frac{f'(\xi)}{g'(\xi)}\to \lim_{t\to a}\frac{f'(t)}{g'(t)}$ . The $\infty/\infty$ case follows from a similar but more delicate estimate.

Topics that use this theorem

Applications of the Derivative

Related theorems

Mean value theorem (Lagrange)

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Guillaume François Antoine de L'Hôpital (1696). Analyse des infiniment petits, pour l'intelligence des lignes courbes
Carl B. Boyer (1968). A History of Mathematics