TITLE: Limit of $x \ln^2|x|$ when $x\to 0$
QUESTION [6 upvotes]: I want to evaluate this limit :$$\lim_{x\to 0}x\ln^2|x|$$ 
I wanted to use L'Hôpital's rule for this:  $\lim\limits_{x\to0-}\frac{\ln^2|x|}{\tfrac{1}{x}}$, but I don't know how to differentiate the  logarithm function, because of the absolute value. 
My other question: do I have to calculate the one sided limits first?

REPLY [2 votes]: You can compute it directly by this substitution:
$$\lim_{x\to0} x\ln^2|x|=\lim_{y\to\infty}\pm e^{-y}\ln^2(e^{-y})=\lim_{y\to\infty}\pm\frac{y^2}{e^y}=0$$