My calculus instructor recently mentioned some odd properties of limits that I don't recall ever seeing, and seem alien to me. He says that the following statements are allowed:
$$\lim_{n\to\infty}\sin\left(\frac{1}{n}\right)=\sin\left(\lim_{n\to\infty}\frac{1}{n}\right)=\sin(0)=0$$
What rules allow for this? The only ones I've been made aware of previously were the following:
$$\lim_{n\to\infty}cf(n)=c\lim_{n\to\infty}f(n)$$
$$\lim_{n\to\infty}f(n)\pm g(n)=\lim_{n\to\infty}f(n)\pm\lim_{n\to\infty}g(n)$$
$$\lim_{n\to\infty}f(n)\cdot g(n)=\lim_{n\to\infty}f(n)\cdot\lim_{n\to\infty}g(n)$$
$$\lim_{n\to\infty}\frac{f(n)}{g(n)}=\frac{\lim_{n\to\infty}f(n)}{\lim_{n\to\infty}g(n)},\lim_{n\to\infty}g(n)\neq 0$$
Best Answer
The general form is $\lim_{n \to \infty} f(a_n) = f(\lim_{n \to \infty}a_n)$.
You can use this if the limit $\lim_{n \to \infty} a_n = a$ exists and $f$ is continuous at $a$. (Which is the case in your example).
In fact, this is one of the equivalent definitions of continuity at a point for real valued functions.