Question: > 
> $\lim\_{x \to \infty} \frac{x}{(R^2+x^2)^{1/2}} $
> 
> 
> 

Attempted applying L'Hopital's Rule to this limit but after two applications I end up up back at the beginning. Am I missing something?

Note: $R$ is a constant here.

Answer: 
Or: first compute
$$
\lim\_{x \to \infty} \frac{x^2}{(R^2+x^2)}
$$
using l'Hopital's rule, then take the square root of your result.
With *equivalents* :

$\bigl(R^2+x^2\bigr)^{1/2}\sim\_\infty \lvert x\rvert $, so
$$\frac{x}{\bigl(R^2+x^2\bigr)^{1/2}}\sim\_\infty \frac{x}{\lvert x\rvert}=\begin{cases}1&\text{if }\,x>0,\\-1&\text{if }\,x<0.\end{cases}$$