Question

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

Answer 1

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]}