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I am a little confused going over some review of limits and sequences.

The question is as follows;

$$\lim_{n\to \infty} n\sin(1/n) $$

What I tried was I rewrote $n\sin(1/n)$ as $$ \frac{\sin(\frac{1}{n})}{\frac{1}{n}} $$

But what the professor did next is where I got a little confused, he introduced $x=\frac{1}{n}$ and then wrote $f(x)=\frac{\sin x}{x}$ and $$\lim_{x\to 0} f(x)= 1 $$ and from this he concluded that the limit of the sequence was $1$. Im not sure how this conclusion was made. Is it always valid to introduce new varaible like that? I feel like I am missing some important theromes relating limits of $n$ and $\frac{1}{n}$.

Thanks in advance for any hints,answers, advice and help.

kingW3
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1 Answers1

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There are different ways of understanding limit os aa function $f$. One way is: $\lim\limits_{x \rightarrow a}f(x)$ exists and equals $b$ if and only if for any sequence $a_n \rightarrow a$ we have that $\lim\limits_{n \rightarrow \infty}f(a_n)$ exists and equals the same number $b$. This is what is used here with $a_n=\dfrac{1}{n}$

voldemort
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