# Properties of limits when dealing with functions and parentheses .

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:

$$\lim_{n\to\infty}cf(n)=c\lim_{n\to\infty}f(n)$$ $$\lim_{n\to\infty}f(n)\pm g(n)=\lim_{n\to\infty}f(n)\pm\lim_{n\to\infty}g(n)$$ $$\lim_{n\to\infty}f(n)\cdot g(n)=\lim_{n\to\infty}f(n)\cdot\lim_{n\to\infty}g(n)$$ $$\lim_{n\to\infty}\frac{f(n)}{g(n)}=\frac{\lim_{n\to\infty}f(n)}{\lim_{n\to\infty}g(n)},\lim_{n\to\infty}g(n)\neq 0$$

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if a function $f$ is continuous at $x_0$, then $\lim_{x\to x_0}f(x)=f(x_0)$ –  yoyo May 27 '13 at 21:02

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.

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OK.. that explanation helps a lot to understand when and where I can actually use such a step. I was afraid that I was using it arbitrarily and in very wrong situations without knowing the reason why it's alowed. –  agent154 May 27 '13 at 21:09

The relation $$\lim_{n\to\infty}\sin\left(\frac{1}{n}\right)=\sin\left(\lim_{n\to\infty}\frac{1}{n}\right)$$ is valid due to the continuity of $\sin$ function.

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What your teacher said is correct. A standard theorem, although more difficult than the four other properties of limits you listed is:

Theorem: If $f$ and $g$ are functions such that $f$ is continuous at $a$ and $g(x)\rightarrow a$ as $x\rightarrow b$ then:

$$\lim_{x\to b} f(g(x))=f\left(\lim_{x\to b} g(x)\right)=f(a).$$

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