**Removable and essential singularities**
A singularity of $f$ at $a$ is

**removable** if $\displaystyle \lim_{x\to a}f(x)$ exists. If the limit exists, then we can extend the domain of $f$ to include $a$ and make $f$ continuous at $a$ by setting $\displaystyle f(a) = \lim_{x\to a}f(x)$, thereby removing the singularity. On the other hand, if $\displaystyle \lim_{x\to a}f(x)$ does not exist, then no possible value for $f(a)$ could make $f$ continuous at $a$, so, in that case, we say that the singularity at $a$ is

**essential.**