$$\lim_{x \rightarrow 0} x \cdot \sin \frac{1}{x}$$
I need to show the work (without looking at a graph) but when I plug in numbers for sin(1/x), I get oscillating values. How do I show algebraically that the function sin(1/x) is oscillating, without plugging in 100 different x-values to show that it oscillates?

Question

$$\lim_{x \rightarrow 0} x \cdot \sin \frac{1}{x}$$
I need to show the work (without looking at a graph) but when I plug in numbers for sin(1/x), I get oscillating values. How do I show algebraically that the function sin(1/x) is oscillating, without plugging in 100 different x-values to show that it oscillates?

anonymous · Answer

It's not very rigourous, but call$$y=\frac{1}{x}$$
$$\sin(y)$$Is obviously oscillating with respect to y, and 1/x (=y) is proportional to x, so there will obviously be some sort of oscillation of sin(y) wrt x.

anonymous · Answer

But then I would also have to show that y = 1/x is oscillating behavior...wouldn't that bring me back to square one?

anonymous · Answer

$$
\lim_{x\to 0}\left |   x \sin\left(\frac 1 x \right)      \right|<\lim_{x\to 0}|x|=0

$$

anonymous · Answer

We us the fact that
$$
| \sin(u)|\le 1

$$

anonymous · Answer

NOT differentiable at 0.