# What is the difference between an asymptote and a hole?

##### 1 Answer
Nov 7, 2015

The two concepts are quite different and only sometimes coincide.

See explanation...

#### Explanation:

A vertical asymptote usually corresponds to a 'hole' in the domain, and a horizontal asymptote often corresponds to a 'hole' in the range, but those are the only correspondences I can think of.

For example, we can define the function $t$ as follows:

$t \left(x\right) = \left\{\begin{matrix}0 & \text{if " x = ((2k+1)pi)/2 " for some " k in ZZ \\ tan(x) & "otherwise}\end{matrix}\right.$

Then $t \left(x\right)$ has vertical asymptotes at $\frac{\left(2 k + 1\right) \pi}{2}$ for all $k \in \mathbb{Z}$, but has no 'holes'.

The function $f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$ has no asymptotes, (unless you count $y = x + 1$), but it has a 'hole' at $x = 1$, where $f \left(x\right)$ is not defined.