$$\lim_{x \rightarrow \infty} x \sin \frac{1}{x}$$
I need to show all my work, but I don't know how to show it. I know sin(1/x) is always between 0 and 1, but sin(1/x) is getting close to 0 as x approaches infinity, and so I'm getting infinity * 0 which doesn't really make any sense.

Question

$$\lim_{x \rightarrow \infty} x \sin \frac{1}{x}$$
I need to show all my work, but I don't know how to show it. I know sin(1/x) is always between 0 and 1, but sin(1/x) is getting close to 0 as x approaches infinity, and so I'm getting infinity * 0 which doesn't really make any sense.

anonymous · Answer

Introduce new variable$$u = \frac{1}{x}$$

anonymous · Answer

Right, thank you

anonymous · Answer

then u have$$\lim_{u\rightarrow 0}\frac{\sin u}{u}$$

anonymous · Answer

which is as well known = 1