# Left/Right Limit Proof

Using a limit from the left and limit from the right argument, how can I see whether a limit exists for the following:

$$\lim_{x \rightarrow 0} \frac{\sin(x)}{|x|}$$

I think this limit does not exist by this argument, but how would I demonstrate that?

• What do you think the limits from the left and right are? – Brian M. Scott Mar 20 '13 at 3:58
• I think -1 from the left and 1 from the right. Is it simple enough to say those are unequal, thus a limit DNE? – Peej Gerard Mar 20 '13 at 4:01
• @PaulGerard Yes. If the limit exists, it has to be unique. – user17762 Mar 20 '13 at 4:02
• Those are right, and yes, it is. The key here is that $\sin x$ is very close to $x$ when $x$ is near $0$, so the fraction is very much like $\dfrac{x}{|x|}$. – Brian M. Scott Mar 20 '13 at 4:02
• ah i see, thanks again! – Peej Gerard Mar 20 '13 at 4:05

$$\lim_{x\to 0^+}\frac{\sin x}{|x|}=\lim_{x\to 0^+}\frac{\sin x}{x}=1$$
$$\lim_{x\to 0^-}\frac{\sin x}{|x|}=\lim_{x\to 0^-}\frac{\sin x}{-x}=-1$$
$$\lim_{x\to 0}\frac{\sin x}{ x}=1$$