If $f(x)$ and $g(x)$ are differentiable functions in $]0,1[$ and it is given that $f$ and $g$ are continuous in $[0,1]$ and we also get that $g(0)=f(0)=0$, is then the following statement, "$\,\lim_{x\rightarrow 0+}\frac{f(x)}{g(x)}= A \implies \lim_{x\rightarrow 0+}\frac{f'(x)}{g'(x)}= A"$, true?
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https://math.stackexchange.com/questions/577597/does-lhôpitals-work-the-other-way – Martin R Nov 26 '21 at 13:36
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Try $f(x)=x$, $g(x)=x+x^2 \sin(1/x)$. – WimC Nov 26 '21 at 15:03
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@WimC yes but g(0)=0, so we get 0+0*sin(1/0) and i don't know if we can assume that this equals zero – Techno Nov 26 '21 at 17:26
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$g$ extends to a continuous function on $[0,1]$. Note that $\lvert g(x) \rvert \leq x+x^2$ on $(0,1)$. – WimC Nov 26 '21 at 17:34