Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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:

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

share|improve this question
2  
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

3 Answers 3

up vote 3 down vote accepted

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.

share|improve this answer
    
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.

share|improve this answer

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:

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.