Does a limit exist at the end point of a function according to the epsilon delta definition?
Andrew Mclaughlin 
According to the epsilon delta definition, the limit L of a function f(x) at a point c in the domain exists if for every $\epsilon > 0$, I can find a $\delta > 0$ such that if $0 < |x - c| < \delta$ then $|f(x) - L| < \epsilon$.
If I have a continuous function defined in $[A, B]$, and I want to calculate $\lim_{x \to A} f(x)$, the I can find the corresponding $\delta$ for for every $\epsilon$. The limit should exist and be equal to $f(A)$.
But the left hand limit of the function doesn't exist, since A is the endpoint, and a limit only exists when the left hand limit is equal to the right hand limit. Therefore the limit should not exist.
Why is it that the limit exists using one definition but not using another? Is there something wrong with how I have interpreted the $\epsilon$ $\delta$ definition?
$\endgroup$ 32 Answers
$\begingroup$A limit in an end point is just the one-sided limit. Only when function is defined on both sides of a given point it is required that both one-sided limits exist and are equal.
There are no problems with the epsilon delta definition, the problem is that you want the left-side limit to exist in the end point when it's not required.
$\endgroup$ 2 $\begingroup$I refer to the answer provided by Adam Latosinski and reiterate the fact that we have the following result:
PropositionIf both limits
$$\lim_{x \to c^+} f(x) \quad \text{and} \quad \lim_{x \to c^-} f(x)$$
exist and coincide, the limit $\lim_{x \to c} f(x)$ exists and equals that common value.
Notice that the existance of both limits is an assumtion in this result. If one of them does not exist, the conclusion may not be true.
Given the quote from your book, I can see how this is confusing. This boils down to definitions. If one is very pedantic, one could take that stance that when a function is not defined on both sides of a point it cannot have a limit at that point, in which case these questions become rather cumbersome. In fact, it is much more convenient to adopt the convention that when a function is not defined to the left of a point and it has a right-sided limit, then that is its limit at that point, and vice versa. So if $f: [a, b] \to \mathbb{R}$ is a function and the limit
$$\lim_{x \to a^+} f(x)$$
exists, we will also write $\lim_{x \to a} f(x)$, where it is understood that the limit is one-sided since the left-handed limit does not make sense.
EDIT: I should also note that some author avoid this problem very elegantly by providing the following definition of a limit: Let $f: D_f \to \mathbb{R}$ a function and $a \in D_f$. We say the $f$ approaches the limit $L$ as $x$ goes to $a$, and write
$$\lim_{x \to a} f(x) = L,$$
if for every $\varepsilon > 0$ there exists $\delta > 0$ such that
$$|L-f(x)| < \varepsilon$$
whenever $x \in D_f$ and $|x-a| < \delta$.
Notice how we require $x$ to be in $D_f$, so the $x$-es outside will not matter.
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