Does L’Hôpital’s work the other way?

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

limxcf(x)g(x)=limxcf(x)g(x)

As

limxcf(x)g(x)=limxcf(x) dxg(x) dx

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

limxcf(x)g(x)=limxcf(x) dxg(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##

(In response to the comments and answers.)

Take 2 functions f and g. When is

limxcf(x)g(x)=limxccxf(a) dacxg(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, 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?

Answer

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

Attribution
Source : Link , Question Author : JMCF125 , Answer Author : Umberto P.

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