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?

  • $\begingroup$ 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

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

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

The above, of course, based in

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.