Question

What kind of functions have horizontal asymptotes?

Answer 1

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(x)={2x+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}{2x+3}/{x^2+1}`

by dividing the numerator and the denominator by `x^2`,

`=lim_{x to pm infty}{2/x+3/x^2}/{1+1/x^2}={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(x)={1+2x-3x^5}/{2x^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}{1+2x-3x^5}/{2x^5+x^4+3}`

by dividing the numerator and the denominator by `x^5`,

`=lim_{x to pm infty}{1/x^5+2/x^4-3}/{2+1/x+3/x^5}={0+0-3}/{2+0+0}=-3/2`,

which means that `y=-3/2` is a horizontal asymptote of `g`.

I hope that this was helpful.