When the limit makes sense

Warning — When the limit makes sense

The limit $\lim_{x\to x_0}f(x)$ is defined only for points $x_0$ that are accumulation points of the domain of $f$. The reason is simple: to “see $f$ approach a value $L$” one needs infinitely many values of $x$ close to $x_0$ (and different from $x_0$); if $x_0$ is isolated, there are none.

An example clarifies the difference between the two cases. The function $f(x) = \sqrt{x-2}$ has domain $D = [2,+\infty)$.

• It makes sense to compute $\lim_{x\to 2}f(x)$, because $2$ is an accumulation point of $D$, even though one cannot “arrive” from the left of $2$: the limit exists only from the right.
• It makes no sense to speak of $\lim_{x\to 0}f(x)$, because $0$ is not an accumulation point of $D$: there exists a neighbourhood of $0$, for example $(-1,1)$, that contains no point of $D$.

Links

Topics: Limits
Concepts: Domain · Limit · Right and left limit · Accumulation point