Home > Calculus > Limits at Infinity and Horizontal Asymptotes

What kind of functions have horizontal asymptotes?

Answer 1

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

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#.

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.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Luke Alvarez

Rational functions have horizontal asymptotes.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.