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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.


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Your teacher clearly does not understand the notion of limit. As $x$ gets closer to $1$ the values of $f(x)$ will get closer to $2$, not $3$. The limit is not affected by what $f$ actually is at $x=1$, it is about what happens as $x$ "approaches" $1$. – fretty Sep 10 '12 at 20:55
up vote 4 down vote accepted

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.

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