Please provide me $f$ and $g$ which satisfy L'Hopital rule's hypotheses but for which $g'$ does not converge.
More precisely, give $f$ and $g$ which are are real and differentiable in $(a,b)$ and $g'(x)\ne0$ for all $x\in(a,b)$, where $-\infty\le a< b\le+\infty.$; and for some $c\in(a,b)$, $$\lim\limits_{x\to c}\frac{f'(x)}{g'(x)}\to A$$ for some real $A$; and $f(x)\to 0$ and $g(x)\to0$ as $x\to c$; and $$\lim\limits_{x\to c}\frac{f(x)}{g(x)}\to A$$, but $$\lim\limits_{x\to c}g'(x)$$ does not exist.
Context: Rudin shows in example 5.18 that if $f:\Bbb R\to \Bbb C$ and $g:\Bbb R\to \Bbb C$ if $g'$ does not converge then L'Hopital may not be applied. I wan't to see an example of real function where $g'$ does not converge and still L'Hopital rule works.
${}{}{}$
All you need a differentiable function $g$ such that $g(c)=0$ and $\lim_{x \to c} g'(x)$ does not exist at some point $c$. You can take $f=g$ in that case so that hypothesis is satisfied with $A=1$.