# Why do some functions have asymptotes?

Sep 2, 2015

Some functions have asymptotes because the denominator equals zero for a particular value of $x$ or because the denominator increases faster than the numerator as $x$ increases.

#### Explanation:

Often, a function $f \left(x\right)$ has a vertical asymptote because its divisor equals zero for some value of $x$.

For example, the function $y = \frac{1}{x}$ exists for every value of $x$ except $x = 0$.

The value of $x$ can get extremely close to $0$, and the value of $y$ will get either a very large positive value or a very large negative value.

So $x = 0$ is a vertical asymptote.

Often a function has a horizontal asymptote because, as $x$ increases, the denominator increases faster than the numerator.

We can see this in the function $y = \frac{1}{x}$ above. The numerator has a constant value of $1$, but as $x$ takes a very large positive or negative value, the value of $y$ gets closer to zero.

So $y = 0$ is a horizontal asymptote.