Using continuity to evaluate limits I hope you guys are enjoying your weekend. I have a question about limits.
This homework problem asks me to use continuity to evaluate this limit, I would like to double-check that I have following the right procedure.
The problem is as follows:
$$\lim_{x\to \pi}\sin(x + \sin x)$$
I break the problem up into two seperate limits:
$$\lim_{x\to \pi}\sin x + \lim_{x\to \pi}\sin(\sin x)$$
Because $\sin x$ is continuous in its domain and its domain includes all real numbers, both limits are continuous and have a domain that includes all real numbers. 
I can therefore plug in $\pi$ and conclude that the limit is $0$.
Is my methodology correct or am I making a mistake?
Thanks!

For a function $f:\mathbf{R}\to\mathbf{R}$ we have that $f$ is continuous at every point in $\mathbf{R}$ if $f$ is "differentiable", that is, able to be differentiated. In this case, if we take $f(x) = \sin(x + \sin x)$, then we have that
$$\frac{\mathrm{d}f}{\mathrm{d}x}=\cos(x+\sin x)(1+\cos x),$$
and so we may justly claim that $f$ is continuous over $\mathbf{R}$. Since 
$$\lim_{x\to a}h(x)=h(a),$$
for some continuous function $h$, we then have
$$\lim_{x\to \pi}\sin(x+\sin x)=\sin(\pi+\sin\pi)=0.$$
