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.
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1$\begingroup$ @Davide: No, it's permissible for $g'(x)=0$ at the point of the limit $\endgroup$– VasiliCommented Mar 8 at 14:22
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1$\begingroup$ 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. $\endgroup$– zkutchCommented Mar 8 at 14:27
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$\begingroup$ The two sides correctly give $+\infty$ as $x \to \pi^-$ and $-\infty$ as $x \to \pi^+$. It is applicable and helpful. $\endgroup$– HenryCommented Mar 9 at 0:58
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2 Answers
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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.
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$\begingroup$ Appreciate your answer and the reference. I see that it would be better to define L'Hospital's rule as one-sided limit from the start as in the Zorich's book. $\endgroup$– VasiliCommented Mar 8 at 14:54
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$\begingroup$ @Vasili : In my opinion it would be better to avoid l'Hôpital at all and just use the extended mean value theorem. This would separate the derivative and limit aspects and thus retains more information. $\endgroup$ Commented Mar 9 at 8:22
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You can use L'Hôpital's rule for one-sided limits. Here you can use it twice, once on each side, and it says the one-sided limits are the same in the first limit as in the second limit (which they are). See Does L'hopital work for one sided limits?
