Limit of sin(1/n)*n

Question

My Maple input limit(sin(1/n)*n,n=infinity); says 1.

I don't understand why $$ \lim_{n \to \infty} \sin\left(\frac{1}{n}\right) \cdot n = 1 $$

I know that $\lim_{n \to \infty} 1/n = 0$, so it kind of says "0 * infinity = 1".

Have I overlooked some rewriting of $\sin(1/n) n$?

Yes. 0*infinity(limit) can equal to one. Remind limit of $\dfrac{1}{n}.n$ — Isura Manchanayake, Dec 06 '15 at 17:40

score 7 · Answer 1 · answered Dec 06 '15 at 17:39

7

Make a change of variables $x={1\over n}$ so that $x\to 0$ as $n\to \infty$, then use $\lim_{x\to 0} {\sin x\over x}=1$.

answered Dec 06 '15 at 17:39

Matt Dickau

2,227

It should be noted that $\lim_{x \to 0} \frac{\sin{x}}{x} = 1$ is a standard limit and is proved here. – Mar 12 '19 at 12:52

score 3 · Answer 2 · edited Dec 06 '15 at 17:49

3

$$n\sin\left(\frac{1}{n}\right)=\frac{\sin\left(\frac{1}{n}\right)}{\frac{1}{n}}$$

edited Dec 06 '15 at 17:49

answered Dec 06 '15 at 17:42

MSE

3,153

E.H.E · Answer 3 · 2015-12-06T17:43:55.077

2

$$\lim_{n \to \infty} \sin\left(\frac{1}{n}\right) \cdot n = \lim_{n \to 0} \frac{\sin n}{n}=1$$

edited Dec 06 '15 at 17:43

answered Dec 06 '15 at 17:39

E.H.E

23,280

You nailed it man. This is as simple as it gets. good use of basics. – Vinay5forPrime Dec 06 '15 at 17:41
1

$n\to 0$ doesnot mean. – R.N Dec 06 '15 at 17:49
$\frac{1}{n} \rightarrow n$ so $\frac{1}{\infty} \rightarrow 0$ – E.H.E Dec 06 '15 at 17:54
you are right if you use for example $x=\frac{1}{n}$, but for natural number we just define limit $n\to\infty$ – R.N Dec 06 '15 at 17:59
I didt write $\frac{1}{n}=n$ but $\frac{1}{n} \rightarrow n$ – E.H.E Dec 06 '15 at 18:05

score 1 · Answer 4 · answered Dec 06 '15 at 17:39

1

Because $$\lim_{x\to 0}\frac{\sin x}{x} = 1,$$which can most easily seen by the Taylor expansion of $\sin x$ about $0,$ $$\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \dots.$$ Taking $x$ as $1/n$ reveals the deep equivalence.

answered Dec 06 '15 at 17:39

CR Drost

2,912

Limit of sin(1/n)*n

4 Answers4