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!

Using continuity to evaluate the problem means that you can use the following fact (assuming you proved it in class, not sure what else your teacher might have been asking) for a continuous function $f:\mathbb{R} \to \mathbb{R}$:
$$\lim_{x\to a} f(x)=f(\lim_{x\to 0} x).$$
So $\lim_{x\to \pi}\sin(x+\sin x)=\sin(\lim_{x\to \pi}x+\sin x)=\sin(\pi+0)=0$
