I had a question about the definition of a limit. I know for a limit to exist the right hand limit must equal the left hand limit but what if the graph of a specific function has the domain from [0,5] and if you wanted to find the limit at x=5 you know the left hand limit exists but the right hand doesn't.

So in this situation does the limit exist or not since only the right-hand limit exists?

  • $\begingroup$ The left hand limit exist only $\endgroup$ – imranfat Mar 29 '16 at 18:42
  • $\begingroup$ No, the limit does exist. $\endgroup$ – D_S Mar 29 '16 at 18:43
  • $\begingroup$ If the right hand limit is not defined, then THE limit does not exist, unless the domain can be extended beyond 5 and both right hand and left hand limit exist and are equal $\endgroup$ – imranfat Mar 29 '16 at 18:44
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    $\begingroup$ Well, only left limit exists (since to the right the function is not defined), but that's ok: the limit coincides with the left limit. $\endgroup$ – Crostul Mar 29 '16 at 18:44
  • $\begingroup$ "The limit coincides with the left limit, so the limit exists? Ehhh...?? $\endgroup$ – imranfat Mar 29 '16 at 18:47

You can only discuss limits in neighborhoods where the function is defined.

If the function is only defined in $[0, 5]$, then it makes no sense to talk about a right-hand limit at $5$. Therefore, when you talk about a limit at $5$, you can only mean left-hand limit.

  • $\begingroup$ That a better way of wording it +1 $\endgroup$ – imranfat Mar 29 '16 at 18:47
  • $\begingroup$ In this case, the left-hand limit and the limit are the same thing though. $\endgroup$ – D_S Mar 29 '16 at 18:58
  • $\begingroup$ So does the Limit exist overall? $\endgroup$ – Asker123 Mar 29 '16 at 19:07
  • $\begingroup$ @Asker123 I think it is better to come up with particular examples... $\endgroup$ – imranfat Mar 29 '16 at 19:13
  • $\begingroup$ For my question above, after reading all of the comments and the answers I think the limit does exist assuming that the limit can exist only on a given domain with boundaries. $\endgroup$ – Asker123 Mar 29 '16 at 19:14

Check out the usual definition of the limit of a function of a real variable (https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit#Precise_statement).

If $D$ is any subset of $\mathbb{R}$, and $f: D \rightarrow \mathbb{R}$ is a function, and $c \in D$ (more generally $c$ can be a limit point of $D$), then we can talk about whether $\lim\limits_{x \to c} f(x)$ exists.

This stuff about left and right hand limits needing to coincide for the limit to exist only applies when $c \in D$ and $D$ contains an open interval about $c$.

For example, suppose $D = [0,5]$, and $f: D \rightarrow \mathbb{R}$ is the function $f(x) = x^2$. Then $\lim\limits_{x \to 5} f(x)$ exists and is equal to $25$, because for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $x \in D$ and $|x - 5| < \delta$, then $|x^2 - 25| < \epsilon$.

So in this case, $\lim\limits_{x \to 5^-} f(x)$ and $\lim\limits_{x \to 5} f(x)$ exist, but it makes no sense to talk about $\lim\limits_{x \to 5^+} f(x)$.


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