Use continuity to evaluate the $\lim_{x\to\pi}(7\sin(x + \sin x))$ I trying to solve the following limit using continuity, but I'm unsure of what using continuity means:
$$\lim _{x→π} 7 \sin(x + \sin x)$$
Also, which strategy should I use to solve questions such as this? If I can just plug $\pi$ into x, I do not see any identities that will help me simplify it any further. 
I am not asking for someone to solve this for me - just a hint or two to get me on my feet would be great. Thanks in advance! 
 A: Continuity of $f$ at $a$ means that $\lim\limits_{x\to a}f(x)=f(a)$. 
In your case $f(x)=7\sin(x+\sin(x))$. Is it continuous at $\pi$?
If it is then the property would allow us to compute $$\lim_{x\to\pi}7\sin(x+\sin(x))=7\sin(\pi+\sin(\pi)).$$
Continuity is not terribly easy to prove directly. But a few properties might help. 
Do we know if $\sin(x)$ is continuous?
How does continuity behave with respect to sum of functions $f(x)+g(x)$, (like $x+\sin(x)$)?
How does is behave with respect to composition $f(g(x))$ (like $\sin(x+\sin(x))$)?
What about multiplying by a constant?
Alternatively, have we heard something about the relationship between elementary functions and continuity? Is $7\sin(x+\sin(x))$ an elementary function?
If we decide, or are forced, to chase the rabbit further into the hole we need to study the continuity of $\sin(x)$ at $x=\pi$. For this we need to show that $|\sin(x)-\sin(\pi)|$ is small when $|x-\pi|$ is small. We can use that $\sin(a)-\sin(b)=2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)$ and that $|\sin(x)|\leq |x|$. From this it follows that 
$$|\sin(x)-\sin(\pi)|=2\left|\cos\left(\frac{x+\pi}{2}\right)\right|\left|\sin\left(\frac{x-\pi}{2}\right)\right|\leq2\left|\sin\left(\frac{x-\pi}{2}\right)\right|\leq|x-\pi|$$
