The Intuition behind l'Hopitals Rule

Question

I understand perfectly well how to apply l'Hopital's rule, and how to prove it, but I've never grokked the theorem. Why is it that we should expect that $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)},$$ but only when specific criteria are met, like $f(x)$ and $g(x)$ approaching $0$ as $x \to a$? I've never found the proof of the theorem to shed much light on this (thought this might be because I don't have much of an intuitive sense as to why the Cauchy Mean Value Theorem holds, either). I've always been mystified as to how l'Hopital (Bernoulli?) ever discovered this theorem, beyond accidentally discovering that it holds in some cases and then generalizing it to a theorem. So, if you have any ideas as to where the theorem came from, and why it makes intuitive sense, I'd appreciate it if you shared them!

Answer 1

There is a case where the result is obvious: Suppose that $f,g$ are continuously differentiable at $a$, that $g'(a)\ne0$, and $f(a)=g(a)=0$. Under these assumptions, $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f(x)-f(a)}{g(x)-g(a)}=\frac{\lim_{x\to a}\frac{f(x)-f(a)}{x-a}}{\lim_{x\to a}\frac{g(x)-g(a)}{x-a}}=\frac{f'(a)}{g'(a)}. $$ Of course, under the assumption that $f',g'$ are continuous at $a$, the latter is also $\displaystyle\lim_{x\to a}\frac{f'(x)}{g'(x)}$. Note that this is just a particular case of the result, but it is one of the most used cases, and it is widely applicable, for example, it also holds of analytic functions on an open domain in the plane, a case definitely not covered by the usual (real valued) argument.

The proof of the theorem is an extrapolation of this case, actually: We note that it is not necessary to assume that $f',g'$ are continuous at $a$, or even that they are defined there, since the differences $f(x)-f(a)$, $g(x)-g(a)$ are related to derivatives via the mean value theorem, so if $f(a)=0=g(a)$, we have $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f(x)-f(a)}{g(x)-g(a)}=\lim_{x\to a}\frac{f'(t_x)}{g'(t_x)} $$ for some $t_x$ between $x$ and $a$, so if $f'(t)/g'(t)$ converges as $t\to a$, then so does the displayed fraction.

The key is really the mean value theorem. Once one sees that this is all is needed in the case $f(a)=g(a)=0$, the other cases can be foreseen, and the argument adapted to cover them as well.

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As you say, l'Hôpital's rule is due to Bernoulli, see here. You may also be interested in these slides by Ádám Besenyei on the history of the mean value theorem. Together with the history of the result, the geometric intuition discussed there may help you find the result more intuitive as well.

Answer 2

The intuition is that although both numerator and denominator tend to zero or infinity, what eventually matters is their respective rate of change. They do not approach zero or infinity at the same rate and thus the one with the highest rate of change dominates the other.

Answer 3

Around $x=a$ each of these functions can be approximated by their tangent line: $$ f(x)\approx f'(a)(x-a)+f(a) $$ $$ g(x)\approx g'(a)(x-a)+g(a). $$ Thus $$ \frac{f(x)}{g(x)}\approx \frac{f'(a)(x-a)+f(a)}{g'(a)(x-a)+g(a)}. $$ Taking the limit of the right hand side gives $\frac{f'(a)}{g'(a)}$.

Answer 4

Suppose $f(x),g(x)\to0$ as $x\to a$. Look at the point $(a,0)$ under a microscope. The graphs of $f$ and $g$ in that view look like straight lines. Recall that $dx$ is an infinitely small increment of $x$, that looks like $0$ unless you use this microscope. The slopes of those lines you see under the microscope are $$ \frac{f(a+dx)-f(a)}{dx} = \frac{f(a+dx)}{dx}\text{ and similarly }\frac{g(a+dx)}{dx}. $$ The ratio of their slopes is therefore $$ \frac{f(a+dx)}{g(a+dx)} $$ which is what in modern notation we call $$ \lim_{x\to a}\frac{f(x)}{g(x)}. $$ But it's the ratio of their slopes, so it's $\displaystyle\lim_{x\to a}\frac{f'(x)}{g'(x)}$.

Answer 5

To simplify @Michael's answer even further, briefly put for infinitesimal $\epsilon$ we have that $f(\epsilon)$ is indistinguishable (in a suitable sense) from $\epsilon f'(\epsilon)$, and similarly for $g$, and therefore $\cfrac{f(\epsilon)}{g(\epsilon)}$ is indistinguishable from $\cfrac{\epsilon f'(\epsilon)}{\epsilon g'(\epsilon)}=\cfrac{f'(\epsilon)}{g'(\epsilon)}$. Since the OP was specifically interested in grokking this could perhaps help.