# Is $\lim_{n\to\infty}x\cdot\cos\left(\frac{1}{n}\right) = 0$?

If I type $$\displaystyle\lim_{n\to\infty}n\cdot \cos\left(\frac{1}{n}\right)$$ in maple, it gives as output 0. Is this correct and if so, why?

I've tried the following myself:

$$\displaystyle\lim_{n\to\infty}n\cdot\cos\left(\frac{1}{n}\right)=\lim_{k\to 0}\frac{\cos k}{k}\ldots$$ ?

Am I on the right track or am I doing something wrong?

What you tried yourself is correct. Also, note that as $$n \to \infty$$, then $$\cos\left(\frac{1}{n}\right) \to \cos(0) = 1$$. Thus, you should be getting
$$\lim_{n\to \infty}n\cos\left(\frac{1}{n}\right) = \infty \tag{1}\label{eq1A}$$
I'm not sure why Maple is giving you a limit of $$0$$. However, if you use a limit of $$n \to 0$$ instead, then since $$-1 \le \cos\left(\frac{1}{n}\right) \le 1$$, you would then get $$0$$. Perhaps you entered the limit value into Maple incorrectly?