Question

How do you find a removable discontinuity for a function?

Answer 1

A discontinuity `a` of a function `f` is removable if `lim_{x to a} f(x)` exists (that is, is a finite number). If the limit fails to exist (for instance, if it is infinite, or there are different one-sided limits, etc), the discontinuity is non-removable.
Thus, to decide if a discontinuity `a` of a function `f` is removable, you need to examine `lim_{x to a} f(x)`.
See this video on "finding discontinuities" for details.