# Is L'Hospital applicable when the limit of derivative quotient does not exist?

According to some sources, i.e. Wiki, the application of L'Hospital's Rule requires four conditions to be satisfied. One of them is the existence of limit of the quotient of the derivatives. We have the following limit: $$\lim_{x\to \pi}\frac{x-\pi}{1+\cos x}=\left[\frac{0}{0}\right]=-\lim_{x\to \pi}\frac{1}{\sin x}$$ The last limit does not exist, but one-sided limits are $$\pm \infty$$. Should we say that the rule is not applicable in this particular case? Or the rule is applicable but not helpful? Or it's applicable and helpful? Note that the original limit also does not exist.

• @Davide: No, it's permissible for $g'(x)=0$ at the point of the limit Commented Mar 8 at 14:22
• L'Hospital requires existence of limits for derivatives, finite or infinite, Vladimir A. Zorich - Mathematical Analysis I-Springer (2016) on pages 248-249, then same be original limit. Commented Mar 8 at 14:27
• The two sides correctly give $+\infty$ as $x \to \pi^-$ and $-\infty$ as $x \to \pi^+$. It is applicable and helpful. Commented Mar 9 at 0:58

Uisng $$1+\cos x= \frac{\sin x}{\tan \frac{x}{2}}$$ we can reduce initial limit to tangent at $$\frac{\pi}{2}$$, which exists one-sided, so L'Hospital gives correct answer and can be call applicable here. In book, mentioned by me in comment, limit is considered one-sided.