# How do you find the asymptotes for #(x^2 + 9) / (9 x - 5 x^2)#?

##### 1 Answer

vertical asymptotes at x = 0 ,

# x =9/5# horizontal asymptote at

# y = -1/5#

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero.

To find the equation of an asymptote let the denominator equal zero.solve:

# 9x - 5x^2 = 0# 'take out common factor' ie x (9 - 5x ) = 0

# rArr x = 0 , x= 9/5 color(black)(" are vertical asymptotes")# Horizontal asymptotes occur as

# lim_(x→±∞) f(x) → 0 # If the degree of the numerator and denominator are equal then the equation can be found by taking the ratio of leading coefficients.

for this function they are equal , both of degree 2

rewriting as :

# (x^2+9)/(-5x^2+9x)# then

# y = 1/(-5) = -1/5 #

horizontal asymptote is# y = -1/5 # Here is the graph of the function as an illustration.

graph{(x^2+9)/(9x-5x^2) [-10, 10, -5, 5]}