Vertical Asymptotes vs. Holes Examples

Find all vertical asymptotes and/or holes of the function 

First we factor:

The denominator has two roots: x = -4 and x = -2. Each of these will provide us with either a hole or a vertical asymptote.

When we simplify f, we find 

Since the root x = -2 is left over after simplification, we have a vertical asymptote at x = -2.

Since the root x = -4 canceled out during simplification, the function will have a hole when x = -4. The y-value of the hole is

Now we know a little bit about what the function will look like:

PICTURE: hole at (-4,4) and vertical asymptote (dashed line) at x = -2.

Find all vertical asymptotes and/or holes of the function 

This factors as

When we simplify, we find

Since -2 is a root of the simplified denominator, we have a vertical asymptote at x = -2. We do not have a hole there, because the term (x + 2) did not cancel out entirely during simplification.

We'll say it again, since it's important: Vertical asymptotes occur at roots (a.k.a. zeros) of the denominator after the rational function has been simplified; holes occur at roots of the denominator that cancel out entirely during the simplification.