L'Hopital's Rule

The L'Hopital's Rule proof I saw back in MATH 147 always seemed a little mysterious to me. You summoned some arbitrary-seeming sequences, invoked the Cauchy Mean Value Theorem and divided in all the right places (then stirred the concoction three times counter-clockwise for good measure). Each step followed logically from the previous one but I never really had a clear idea of the bigger picture. The sequence-free squeeze theorem-based proof on the wikipedia page didn't do much better for me, either.

I was working on something unrelated yesterday when the following insight popped into my head. It's a tidy little geometric interpretation of the $0/0$ case that gives a nice sort of context to the proofs I've seen.

L'Hopital's Rule

Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$ that are differentiable on the interval $I = [a-\delta, a+\delta]$ except possibly at $a$. If

  • $\lim_{x \to a} f(x) = \lim_{x \to a} g(x)$ and $\lim_{x \to a} f(x) = 0$ or $\infty$, and

  • $\lim_{x \to a} \frac{f'(x)}{g'(x)} = L$ for some $L \in \mathbb{R}$,

then $\lim_{x \to a} \frac{f(x)}{g(x)} = L$.

Consider the $0/0$ case. Without loss of generality, we can (re)define $f(a)$ and $g(a)$ to be $0$ without affecting the behaviour of the limits $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$. We can then use the fact that $f$ and $g$ are continuous at $a$.

Since $f$ and $g$ are differentiable arbitrarily close to $a$, for $x \neq a$ close enough to $a$, we have $f(x) \approx f^\prime(a)*(x-a) + f(a)$ and $g(x) \approx g^\prime(a)*(x-a) + g(a)$.

$$\lim_{x \to a} \frac{f(x)}{g(x)} \approx \lim_{x \to a} \frac{f^\prime(a)*(x-a) + f(a)}{g^\prime(a)*(x-a) + g(a)}$$

But recall that $f(a) = g(a) = 0$. We then get

$$\lim_{x \to a} \frac{f(x)}{g(x)} \approx \lim_{x \to a} \frac{f^\prime(a)*(x-a)}{g^\prime(a)*(x-a)} \approx \lim_{x \to a} \frac{f^\prime(a)}{g^\prime(a)} $$

because we can cancel the pair of $(x-a)$s.

This is touched on more formally in the wikipedia article using the squeeze theorem with upper and lower bounds on the ratio between $f^\prime(x)$ and $g^\prime(x)$ in the interval $I$ (recall that we can't actually use the linear approximation of $f$ at $a$ outright because we don't actually know that $f^\prime(a)$ and $g^\prime(a)$ exist). The proof in the wiki article takes advantage of the continuity of $f$ and $g$ to squish in something like a slope of a linear approximation of $f$ at $a$. The Cauchy Mean Value Theorem is used to carefully sidestep the problems that would result if $g^\prime(x) = 0$.

At the heart of it though, L'Hopital's rule just seems to be a marriage of the ideas that differentiable functions are pretty darn close to their linear approximations at some point as long as you don't stray too far from that point and that for a continuous function, a small movement in the domain means a small movement in the value of the function.