TITLE: Circular uses of L’Hôpital’s rule
QUESTION [15 upvotes]: If you try to find the $\lim_{x\to \infty}\frac{\sqrt{x^2+1}}{x}$
using L’Hôpital’s rule, you’ll find that it flip-flops back and forth between $\frac{\sqrt{x^2+1}}{x}$ and $\frac{x}{\sqrt{x^2+1}}$.
Are there other expressions that do a similar thing when L’Hôpital’s rule is applied to them? I already know that this applies to any fraction of the form $\frac{\sqrt{x^{2n}+c}}{x^n}$.

REPLY [7 votes]: Here is a list of limits related to the OP question. L'Hospital's rule cannot be used for each. 

$\lim\limits_{x\rightarrow 0}\dfrac{x^{2}\sin \left( \frac{1}{x}%
\right) }{\sin x}$
$\lim\limits_{x\rightarrow \infty }\dfrac{x-\sin x}{x+\sin x}$
$\lim\limits_{x\rightarrow \infty }\dfrac{\left( 2x+\sin 2x\right) }{%
\left( 2x\sin x\right) e^{\sin x}}$
$\lim\limits_{x\rightarrow \infty }\dfrac{\sqrt{2+x^{2}}}{x}$
$\lim\limits_{x\rightarrow 0}\dfrac{1-\cos x}{\cos x}$
$\lim\limits_{x\rightarrow \infty }\dfrac{\sqrt{9x+1}}{\sqrt{x+1}}$
$\lim\limits_{x\rightarrow 0}\dfrac{\sqrt{x}}{\sqrt{\sin x}}$
$\lim\limits_{x\rightarrow 0}\dfrac{\cot x}{\csc x}$
$\lim\limits_{x\rightarrow \left( \pi /2\right) ^{-}}\dfrac{\sec x}{%
\tan x}.$