TITLE: How many times do you have to use L'Hôpital's rule?
QUESTION [25 upvotes]: I'm looking for cases like $$\lim_{x \to 0} \frac {1-\cos(x)}{x^2}$$ that will not give you the answer the first time you use L'Hôpital's rule on them. For example in this case it will result in a number $\frac{1}{2}$ the second time you use L'Hôpital's rule. I want examples of limits like  $\lim_{x \to c} \frac {f(x)}{g(x)}$ so that you have to use L'Hôpital's rule $5$ times, $18$  times, or say $n$ times on them to get an answer. Another question is about the case in which you use L'Hôpital's rule as many times as you want but you always end with $\lim_{x \to 0} \frac {0}{0}$. Does this case exist?

REPLY [8 votes]: For the "$\frac{\infty}{\infty}$" case, if you only use the L'Hôpital's rule and don't change your fraction between its successive applications, then one of simple nontrivial never-ending examples from many textbooks is
$$\lim_{x\to0+}\frac{\ln x}{\cot x}$$.