How can i find horizontal asymptote?

Question

Christopher Allers · Accepted Answer

The horizontal line #y=b# is called a horizontal asymptote of #f(x)# if either #lim_{x to +infty}f(x)=b# or #lim_{x to -infty}f(x)=b#.  In order to find horizontal asymptotes, you need to evaluate limits at infinity. Let us find horizontal asymptotes of #f(x)={2x^2}/{1-3x^2}#.
Since
#lim_{x to +infty}{2x^2}/{1-3x^2}=lim_{x to +infty}{2x^2}/{1-3x^2}cdot{1/x^2}/{1/x^2}
=lim_{x to +infty}{2}/{1/x^2-3}=2/{0-3}=-2/3# 
and
#lim_{x to -infty}{2x^2}/{1-3x^2}=lim_{x to -infty}{2x^2}/{1-3x^2}cdot{1/x^2}/{1/x^2}
=lim_{x to -infty}{2}/{1/x^2-3}=2/{0-3}=-2/3#,
#y=-2/3# is the only horizontal asymptote of #f(x)#. (Note: In this example, there is only one horizontal asymptote since the above two limits happen to be the same, but there could be at most two horizontal asymptotes in general.)

Cameron Alexander · Answer

undefined To find the horizontal asymptote of a function, follow these steps:

1. Determine the degree of the numerator and denominator of the function.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
3. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of both the numerator and denominator to find the horizontal asymptote.
4. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.