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Imagine we have a point $a$ of the domain $D$ of function $f$.

The definition of limit I'm using is the following:

$$\lim_{x\rightarrow a}f(x)=b \Leftrightarrow \forall_{\epsilon}\exists_{\delta}\forall_{x}(x\in D \ \land \ x\in N_{\delta}(a)\ \Rightarrow \ f(x) \in N_{\epsilon}(b) )$$, where $N_{\epsilon}(b)$ is the neighbourhood of length $2\epsilon$ at point $b$.

If I pick a point $a$ outside the domain of $f$, then the implication is vacuously true, for any point $b$... Then how can I say that there's no limit?

An old man in the sea.
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1 Answers1

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The exact definition of the limit is as follows:

Let $D\subseteq \mathbb R$ and let $f:D\to \mathbb R$ be a function. Let $a$ be a limit point of $D$. Then, the funciton $f$ has a limit of $L$ at $a$ if and only if for every $\epsilon$, there exists a $\delta$ such that for all $x\in D$, $|x-a|<\delta$ implies $|f(x)-L|<\epsilon$.

If you take a point "far" outside the domain of $f$, you cannot talk about limits.

In early calculus, you often demand even more of $f$, specifically that the limit is only defined if $f$ is defined on some interval $(a,a+\epsilon)$ or $(a-\epsilon, a)$ (or both).

In any case, taking a point "too far" from the domain means that the limit does not exist by definition.

5xum
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  • Could you show that, from the definition? – An old man in the sea. Aug 30 '16 at 12:09
  • @Anoldmaninthesea. It's in the definition. Your definition of limit is incomplete, because the definition needs to demand that $f$ is defined on some (half-) neighborhood of $f$. More specifically, $a$ must be an element of the closure of $D$ by definition. So, if $f$ is defined on $(0,1)$, the limit of $f$ as $x\to -1$ does not exist by definition. – 5xum Aug 30 '16 at 12:09
  • ah... $a$ needs to be an adherent point of $D$. Ok. Thanks I got it now. ;) – An old man in the sea. Aug 30 '16 at 12:13
  • 5xum, I'm not sure you demand to be a limit point... Otherwise, we wouldn't continuity at isolated points of the Domain http://math.stackexchange.com/questions/626089/continuity-of-the-function/626115#626115 – An old man in the sea. Aug 30 '16 at 12:23
  • @Anoldmaninthesea. Well, my textbook does demand the point to be a limit point... – 5xum Aug 30 '16 at 12:28