# What kind of functions have horizontal asymptotes?

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.