I am having some confusion about holes in rational functions. As I'm aware, a hole is where both the numerator and denominator become zero due to some discontinuity. For example,
f(x) = (x+1)(x-1)/(x+1)
would have a hole at x = -1.
What is the point of distinguishing between a hole and Vertical Asymptote?
What is the logic behind the numerator needing to having a discontinuity at the same point as the denominator?
Best Answer
Both the numerator and the denominator being zero is a necessary but not sufficient condition for a hole; see for example the function $f(x) = \frac{x+1}{(x+1)^2}$. The difference between a hole and a vertical asymptote is that the function doesn't become infinite at a hole. See it for yourself: Take your $f$, evaluate it at points close to $x=-1$ and compare to what happens with mine.