The horizontal asymptote of a rational function tells us the limiting value of that function as it approaches infinity.
For a rational function to have a horizontal asymptote of
[tex] \frac{2}{9} [/tex]
then,[tex]<b>the highest degree of x in the numerator must be equal to the highest degree of x in the denominator.</b>[/tex]
The second condition is that,
[tex]<b>the coefficient of the highest degree in the numerator and the coefficient of the highest degree in the denominator should be in the ratio 2:9.</b>[/tex]
Example are given in the graph above.
Here are some other examples,
[tex] y = \frac{2x - 1}{9x + 2} [/tex]
[tex]y = \frac{1 - 6 {x}^{3} }{5x - 27 {x}^{3} } [/tex]
