0
$\begingroup$

If we have a function of real-variable, then we can talk about one-sided limits $\lim\limits_{x\to a^+} f(x)$ and $\lim\limits_{x\to a^-} f(x)$ and, of course, also about the limit $\lim\limits_{x\to a} f(x)$.

I have noticed in the Wikipedia article on one-sided limits (here is link to the current revision) the following claim:

The two one-sided limits exist and are equal if the limit of $f(x)$ as $x$' approaches $a$ exists. In some cases in which the limit $$\lim_{x\to a} f(x)$$ does not exist, the two one-sided limits nonetheless exist. Consequently the limit as $x$ approaches $a$ is sometimes called a "two-sided limit".

This does not seem correct to me. See for example this post: Question Regarding Existence of One Sided Limits

But I want to ask anyway: May this discrepancy be caused by using different definition of limit? (I.e., is there a definition of a limit where it indeed makes sense differentiating between limit and two-sided limit?) Or is the Wikipedia article indeed correct and I missed something obvious?


EDIT:

It seems that the only problem was me misunderstanding the wording in Wikipedia. I understood the formulation in the above paragraph as indication that it is possible that it is possible two-sided limit exists but limit at a does not exist.

Daniel Fischer's comment below is probably the correct interpretation:

It is sometimes called a two-sided limit to emphasise that it is not a one-sided limit that is spoken about. That's how I understand the sentence in wikipedia.

$\endgroup$
  • 2
    $\begingroup$ It is sometimes called a two-sided limit to emphasise that it is not a one-sided limit that is spoken about. That's how I understand the sentence in wikipedia. $\endgroup$ – Daniel Fischer Apr 18 '14 at 7:14
  • $\begingroup$ @DanielFischer It seems that I did not read the article carefully enough. I understood that sentence as indication that it is possible that it is possible two-sided limit exists but limit at $a$ does not exist. If you make your comment into an answer, I will accept it. $\endgroup$ – Martin Sleziak Apr 18 '14 at 7:17
2
$\begingroup$

It's very much situational take the example

$f(x)= \left\{ \begin{array}{ll} 2 & \mbox{if $x \lt 0$};\\ 1 & \mbox{if $x \ge 0$}.\end{array} \right.$

Here both the left and right limits exist, the left is 2, and the right limit is 1, they are not equal so the limit does not exist.

This is because the existence of the left and right limits are a necessary but not a sufficient condition for the limit to exist.

I.e existence of the limit $\Rightarrow$ left and right limit exist.

But the other way does not necessarily hold.

$\endgroup$
  • $\begingroup$ What I consider the usual definition of limit is this one: $\forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon)$. So I suppose that your definition does not require $0<|x-c|$. (I.e., your definition requires the values being close to the limit on some neighborhood of $a$, not only in some punctured neighborhood.) IIRC both these definitions are used in literature. $\endgroup$ – Martin Sleziak Apr 18 '14 at 7:23
  • $\begingroup$ Sorry yeah I meant 1 $\endgroup$ – Ellya Apr 18 '14 at 7:33
  • $\begingroup$ I have edited, I got mixed up with continuity. $\endgroup$ – Ellya Apr 18 '14 at 7:45
1
$\begingroup$

I think that in some cases we have $\lim\limits_{x\to a^{+}} f(x)$ and $\lim\limits_{x\to a } f(x)$ exist but we do not concern about $\lim\limits_{x\to a^{-}} f(x)$. Infact, if $f(x)=\sqrt{x}$ and $a=0$, then $\lim\limits_{x\to 0^{+}} \sqrt{x}=\lim\limits_{x\to 0} \sqrt{x}=0$ but the left hand side limit is undefined.

$\endgroup$

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.