Solving the Limit of (1-cos x)sin(1/x)

[nietzsche]

Homework Statement

Find the following limit:

$$\lim_{x \to 0} (1-\text{cos }x)\text{sin }\frac{1}{x}$$

Homework Equations

The Attempt at a Solution

(1-cos x) -> 0 as x -> 0. sin (1/x) oscillates infinitely many times as x -> 0.

intuition tells me that the limit is 0, but how do i show that?

some ideas i have are using the fact that |sin(1/x)| =< 1, but I'm not sure.

 

[zcd]

Try the squeeze theorem with something that converges to zero like $$\frac{1-\cos{x}}{x}$$.

 

[nietzsche]

i ended up doing this.

$$\begin{align*}<br /> -1 &\leq& \text{sin }\frac{1}{x} &\leq& 1\\<br /> -(1-\text{cos }x) &\leq& (1-\text{cos }x)(\text{sin }\frac{1}{x}) &\leq& 1- \text{cos }x<br /> \end{align*}$$

since both of the terms on the side equal 0 at x=0, by the squeeze theorem, the middle term also goes to 0.

 

[Dick]

That's how I would have done it. Well done!