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Take the function $f(x) = x$, but the furthest defined point to the right on the graph is $x=2$. After that, the line of the graph simply terminates.

Does $\lim_{x\rightarrow 2}$ exist in this situation? $x$ clearly approaches $2$ in the graph, but it only approaches from the left. It doesn't approach from the right since there is no line of the graph there. Building on this, if $\lim_{x\rightarrow 2}$ does not exist, then I suppose the function is not continuous at $x=2$?

This seems like an elementary question, so apologies if this has been asked elsewhere (I searched around but couldn't find a matching question).

Ѕᴀᴀᴅ
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Inertial Ignorance
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  • When we talk about function, we need to specify the domain where this function is defined on. In our case $f(x)=x$ is defined on, say $[0,2]$, since it was not specified what it would be when $x>2$. In this case we can still speak of one sided limit, where we only look at what value $f$ is approaching to when x is approaching from the left. – lEm Oct 04 '18 at 06:09
  • So the only reason we generally look at the left-sided and right-sided limits of a function is because the function's domain is usually defined on both sides? If it's not, then there's no reason to look at both sides? – Inertial Ignorance Oct 04 '18 at 06:11
  • It is better to think of it this way: If the function is not defined on one of the sides, then we cannot even talk about the usually two-sided limit. The reason we look at two-sided limit is because the notion is related to continuous functions. – lEm Oct 04 '18 at 06:14
  • @lEm So does this imply that the limit can exist if f(x) is only defined on one side of x=a, but f(x) is not continuous at x=a? – Inertial Ignorance Oct 04 '18 at 06:26
  • Related: https://math.stackexchange.com/questions/1545193/reconciling-rudins-limit-definitions-one-sided-limit-uniqueness, https://math.stackexchange.com/questions/264609/if-there-is-only-one-one-sided-limit-the-limit-exists – Hans Lundmark Oct 04 '18 at 07:34

1 Answers1

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From https://en.m.wikipedia.org/wiki/Limit_of_a_function

More general subsets

Apart from open intervals, limits can be defined for functions on arbitrary subsets of $\mathbb{R}$, as follows. Let $f$ be a real-valued function defined on a subset $S$ of the real line. Let $p$ be a limit point of $S$—that is, $p$ is the limit of some sequence of elements of $S$ distinct from $p$. The limit of $f$, as $x$ approaches $p$ from values in $S$, is $L$ if, for every $ε > 0$, there exists a $δ > 0$ such that $0 < |x − p| < δ$ and(!) $x ∈ S$ implies $|f(x) − L| < ε$.

This limit is often written

$\displaystyle L={\underset {x\in S}{\lim _{x\to p}}}f(x)$.

The condition that f be defined on S is that S be a subset of the domain of f. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking S to be an open interval of the form $\displaystyle (-\infty ,a)$), and right-handed limits (e.g., by taking S to be an open interval of the form $\displaystyle (a,\infty )$). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the Square root function f(x)=√x can have limit 0 as x approaches 0 from above. (End of citation from Wikipedia)

Conclusion: The definition of Limits heavily depends in the domain of definition for the function to be investigate.

Jens Schwaiger
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