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I know that L'Hospital's rule is applied when $\lim \frac{f'(x)}{g'(x)}$ must exist.

Then, is there an example that $\lim \frac{f'(x)}{g'(x)}$ does not exist but other conditions of L'Hospital's rule hold ?

i.e. for example) Are there functions $f$ and $g$ such that

$\lim f(x) = \lim g(x) = + \infty$ and $f$, $g$ are differentiable but $\lim \frac{f'(x)}{g'(x)}$ does not exist?

I don't speak english very well. I'm Sorry if you don't understand.

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We can use $f(x)=x+\sin x$, $g(x)=x$. In that case, the limit as $x\to\infty$ of the ratio exists, but the limit of the ratio of the derivatives does not.

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If lim f' (x) / g' (x) doesn't exist, then l'Hospital's rule just doesn't give you any information. It doesn't say anything, neither positive nor negative, about the limit of f (x) / g (x).

It is a simple statement of the form "if A and B and C then D". If either of A or B or C isn't true, then the rule is trivially true, but doesn't help in any way.

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