How to find the limit of $\sin(f(x))$, given the graph of $f(x)$

Question

The full question is uploaded here: https://i.stack.imgur.com/zwIlv.jpg

Basically, given the graph shown in the image above, I thought that the limit of $\sin(f(x))$ would be $\sin(2)$, since the limit of just $f(x)$ is $2$. However, my teacher says that it is $\sin(3)$, since $f(x) = 3$ at that point. Can anyone explain why it is one and not the other?

Also, if you could provide a source, that would be great, so I can show my teacher if I am right.

Thanks.

Answer 1

You are correct, your teacher is not. Since $\lim\limits_{x\to 1}f(x)=2$, we have $$\lim\limits_{x\to 1}\sin(f(x))=\sin\left(\lim\limits_{x\to 1}f(x)\right)=\sin(2)$$ as in general we have $\lim\limits_{x\to a}g(f(x))=g(\lim\limits_{x\to a}f(x))$ whenever $g$ is continuous and $\lim\limits_{x\to a}f(x)$ exists.

Answer 2

You are right, it should be sin(2), I think because of radian and degree mode. If you set the calculator to radian mode, sin(2) = 0.909

I don't know why this questions is in radian mode, but in general you should set your calculator in degree moden

So the answer is A