Can I apply L'Hôpital to $\lim_{x \to \infty} \frac{x+\ln x}{x-\ln x}$? Can I apply L'Hôpital to this limit: 
$$\lim_{x \to \infty} \frac{x+\ln x}{x-\ln x}?$$
I am not sure if I can because I learnt that I use L'Hôpital only if we have $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and here $x-\ln x$ is $\infty-\infty$ and x tends to infinity.

$x -\ln x$ goes to $+\infty$ if and only if $e^{x-\ln x}$ does. And this is the case, since
$$e^{x-\ln x} = \frac{e^x}{x} \ge \frac{1+x+\dfrac{x^2}{2}}{x} \to +\infty $$
as $x \to +\infty$. So yes, you can apply de l'Hopital from the beginning.
