= Derivative quotient limits on partial domains
The usual zero-over-zero <L'Hopital rule> requires the denominator's <derivative> to be nonzero throughout a sufficiently small deleted neighborhood. A limit taken only where that <derivative> is nonzero is insufficient. An explicit counterexample uses $x_n=2^{-n}$, $h_n=4^{-n}$, $S(u)=3u^2-2u^3$ and $q(u)=16u^2(1-u)^2$. On $[3x_n/4,x_n]$ put $g=h_n$ and $f=h_nq((x-3x_n/4)/(x_n/4))$. On $[x_n/2,3x_n/4]$ put $g=h_{n+1}+(h_n-h_{n+1})S((x-x_n/2)/(x_n/4))$ and $f=0$. Extend $g=1,f=0$ beyond one. Values and first <derivatives> agree at the joins, both functions tend to zero, and $g$ is positive. Where $g'\ne0$, it lies inside a transition interval and $f'=0$. Thus $f'/g'$ has natural-domain limit zero, but $f/g$ equals zero at $x_n$ and one at $7x_n/8$, so has no limit.
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