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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?

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  • $\begingroup$ What do you think the limits from the left and right are? $\endgroup$ – Brian M. Scott Mar 20 '13 at 3:58
  • $\begingroup$ I think -1 from the left and 1 from the right. Is it simple enough to say those are unequal, thus a limit DNE? $\endgroup$ – Peej Gerard Mar 20 '13 at 4:01
  • $\begingroup$ @PaulGerard Yes. If the limit exists, it has to be unique. $\endgroup$ – user17762 Mar 20 '13 at 4:02
  • $\begingroup$ 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 $\dfrac{x}{|x|}$. $\endgroup$ – Brian M. Scott Mar 20 '13 at 4:02
  • $\begingroup$ ah i see, thanks again! $\endgroup$ – Peej Gerard Mar 20 '13 at 4:05
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$$\lim_{x\to 0^+}\frac{\sin x}{|x|}=\lim_{x\to 0^+}\frac{\sin x}{x}=1$$

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

The above, of course, based in

$$\lim_{x\to 0}\frac{\sin x}{ x}=1$$

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