We now give the definition of the limit of a function at a boundary
point of its domain. This definition is consistent with limits at
boundary points of regions in the plane and in space, as we will see
in Chapter 14. When the domain of $ƒ$ is an interval lying to the left
of $c$, such as $(a, c]$ or $(a, c)$, then we say that $ƒ$ has a limit
at c if it has a left-hand limit at $c$. Similarly, if the domain of
$ƒ$ is an interval lying to the right of c, such as $[c, b)$ or $(c,
> b)$, then we say that $ƒ$ has a limit at $c$ if it has a right-hand
limit at $c$.Consider the function $f(x)=\sqrt{4-x^2}$ with domain $[-2, 2]$. This
function has a two-sided limit at each point in $(-2, 2)$. It has a
left-hand limit at $x = 2$ and a right-hand limit at $x = -2$. The
function does not have a left-hand limit at $x = -2$ or a right-hand
limit at $x = 2$. It does not have a two-sided limit at either $-2$ or
$2$ because $ƒ$ is not defined on both sides of these points. At the
domain boundary points, where the domain is an interval on one side of
the point, we have $ \lim_{x \to -2} \sqrt{4-x^2} = 0$ and $ \lim_{x\to 2} \sqrt{4-x^2} = 0$.
The function $ƒ$ does have a limit at $x =-2$
and at $x = 2$.
This is from my book. There are some things that is still vague to me.
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Is it true that $ \lim_{x\to 2} f(x) = 0$ and $ \lim_{x \to -2} f(x) = 0$ implicitly means that the left/right limit exists, and not both sides? Does saying $f$ is continuous at $x=-2$ and $x=2$ means that $f$ is implicitly right continuous (for $x=-2$ only) and left continous (for $x=2$ only), and not both?
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How about if I have a function $a(x)=\sqrt{4-x^2}$ for all $x \in [-2,2]$, $a(x)=0$ for all $x<-2$ or $x>2$. Does it still apply $ \lim_{x \to -2} a(x) = 0$ and $ \lim_{x \to 2} a(x) = 0$, however, now it's implicitly meant from both sides?
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If $b(x)=\sqrt{4-x^2}$ for all $x \in [-2,2]$, $b(-3)=0$. Is it still true that $ \lim_{x \to -2} b(x) = 0$, I don't think so since the domain is not $[-2,2]$, but please verify. How about the continuity of $b$ at $x=-2$?
Best Answer
Inside the domain, you have
a limit if both the left- and right- limits exist and are equal;
at the same time, it is a two-sided limit.
At an endpoint,
a limit if the left- or right- limit exists;
there is no two-sided limit.