 L'Hôpital's rule

A standard one-sided version is as follows. Let $f,g$ be differentiable on $(a,b)$, with $g^{\prime }(x)\ne 0$ there. Suppose either $f(x),g(x)\to 0$ as $x\downarrow a$, or $|g(x)|\to \infty$ there. Ifthen $f(x)/g(x)\to L$. The denominator is nonzero sufficiently near the endpoint. Corresponding versions apply at other finite endpoints and at infinity. The quotient of derivatives must have a limit; differentiating numerator and denominator is not an unrestricted quotient identity.

In the zero-over-zero case, extend both functions continuously to the endpoint by zero and use the Cauchy mean value theorem to write $f(x)/g(x)=f^{\prime }({\xi }_x)/g^{\prime }({\xi }_x)$ for an intermediate ${\xi }_x$. For example, $\sin x/x\to 1$ as $x\to 0$. The rule is sufficient, not necessary: $(x+\sin x)/x\to 1$ as $x\to \infty$, while its derivative ratio $1+\cos x$ has no limit. Without the required indeterminate setting, it can give a wrong answer: $(x+1)/x$ diverges at zero although its derivative ratio is one.