When the limit makes sense

Statement

For each function, state whether it makes sense to take the limit at the specified point:

• (a) $f(x) = \ln(x^2-4)$ at $x_0 = 2$;
• (b) $f(x) = \dfrac{1}{x-1}$ at $x_0 = 1$;
• (c) $f(x) = \sqrt{1-x^2}$ at $x_0 = 2$;
• (d) $f(x) = \tan x$ at $x_0 = \pi/2$.

Solution

It makes sense to take the limit at $x_0$ if and only if $x_0$ is an accumulation point of the domain.

• (a) Domain: $x^2-4>0$, i.e. $x<-2$ or $x>2$. The point $2$ is an accumulation point (to the right there are points of the domain arbitrarily close): it makes sense (limit from the right).
• (b) Domain: $\mathbb{R}\setminus\{1\}$. The point $1$ is an accumulation point: it makes sense.
• (c) Domain: $1-x^2\ge 0$, i.e. $[-1,1]$. The point $2$ is not an accumulation point (the neighbourhood $(1{,}5;\ 2{,}5)$ contains no points of the domain): it does not make sense.
• (d) Domain: $\mathbb{R}\setminus\{\pi/2+k\pi\}$. The point $\pi/2$ is excluded from the domain but is an accumulation point of it: it makes sense.

Links

Topics: Limits
Concepts: Domain · Accumulation point
Skills: Reasoning by cases
Exercise type: True or false