# What kind of functions have horizontal asymptotes?

##### 1 Answer
Oct 10, 2014

In most cases, there are two types of functions that have horizontal asymptotes.

1. Functions in quotient form whose denominators are bigger than numerators when $x$ is large positive or large negative.

ex.) $f \left(x\right) = \frac{2 x + 3}{{x}^{2} + 1}$

(As you can see, the numerator is a linear function grows much slower than the denominator, which is a quadratic function.)

${\lim}_{x \to \pm \infty} \frac{2 x + 3}{{x}^{2} + 1}$

by dividing the numerator and the denominator by ${x}^{2}$,

$= {\lim}_{x \to \pm \infty} \frac{\frac{2}{x} + \frac{3}{x} ^ 2}{1 + \frac{1}{x} ^ 2} = \frac{0 + 0}{1 + 0} = 0$,

which means that $y = 0$ is a horizontal asymptote of $f$.

1. Function in quotient form whose numerators and denominators are comparable in growth rates.

ex.) $g \left(x\right) = \frac{1 + 2 x - 3 {x}^{5}}{2 {x}^{5} + {x}^{4} + 3}$

(As you can see, the numerator and the denominator are both polynomial of degree 5, so their growth rates are very similar.)

${\lim}_{x \to \pm \infty} \frac{1 + 2 x - 3 {x}^{5}}{2 {x}^{5} + {x}^{4} + 3}$

by dividing the numerator and the denominator by ${x}^{5}$,

$= {\lim}_{x \to \pm \infty} \frac{\frac{1}{x} ^ 5 + \frac{2}{x} ^ 4 - 3}{2 + \frac{1}{x} + \frac{3}{x} ^ 5} = \frac{0 + 0 - 3}{2 + 0 + 0} = - \frac{3}{2}$,

which means that $y = - \frac{3}{2}$ is a horizontal asymptote of $g$.

I hope that this was helpful.