TITLE: Calculus Question: $\lim_{x \rightarrow 0}\sin(x)\ln{\sin{x}}$
QUESTION [6 upvotes]: I'm having trouble to compute
$$\lim_{x \rightarrow 0}\sin(x)\ln{\sin{x}}$$
I tried to find this limit using Wolfram Alpha and the result is 0, but I don't know how to get this 0.

REPLY [4 votes]: Make a substitution.  Setting $u = \sin x$ (and noting the continuity of $\sin x$), we have
$$
\lim_{x \to 0} \sin(x)\ln(\sin(x)) = \lim_{u \to \sin(0)}u \,\ln u
= \lim_{u \to 0}u \,\ln u
$$
This is a standard problem usually done using L'Hôpital's rule.