# Does L’Hôpital’s work the other way?

As referred in Wikipedia (see the specified criteria there), L’Hôpital’s rule says,

$$limx→cf(x)g(x)=limx→cf′(x)g′(x) \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}$$

As

$$limx→cf′(x)g′(x)=limx→c∫f′(x) dx∫g′(x) dx \lim_{x\to c}\frac{f'(x)}{g'(x)}= \lim_{x\to c}\frac{\int f'(x)\ dx}{\int g'(x)\ dx}$$

Just out of curiosity, can you integrate instead of taking a derivative?
Does

$$limx→cf(x)g(x)=limx→c∫f(x) dx∫g(x) dx \lim_{x\to c}\frac{f(x)}{g(x)}= \lim_{x\to c}\frac{\int f(x)\ dx}{\int g(x)\ dx}$$

work? (given the specifications in Wikipedia only the other way around: the function must be integrable by some method, etc.) When? Would it have any practical use? I hope this doesn’t sound stupid, it just occurred to me, and I can’t find the answer myself.

##Edit##

Take 2 functions $$ff$$ and $$gg$$. When is

$$limx→cf(x)g(x)=limx→c∫cxf(a) da∫cxg(a) da \lim_{x\to c}\frac{f(x)}{g(x)}= \lim_{x\to c}\frac{\int_x^c f(a)\ da}{\int_x^c g(a)\ da}$$

true?

Not saying that it always works, however, it sometimes may help. Sometimes one can apply l’Hôpital’s even when an indefinite form isn’t reached. Maybe this only works on exceptional cases.

Most functions are simplified by taking their derivative, but it may happen by integration as well (say $$∫1x2 dx=−1x+C\int \frac1{x^2}\ dx=-\frac1x+C$$, that is simpler). In a few of those cases, integrating functions of both nominator and denominator may simplify.

What do those (hypothetical) functions have to make it work? And even in those cases, is is ever useful? How? Why/why not?

With L’Hospital’s rule your limit must be of the form $\dfrac 00$, so your antiderivatives must take the value $0$ at $c$. In this case you have provided $g$ satisfies the usual hypothesis that $g(x) \not= 0$ in a deleted neighborhood of $c$.