Question

How do you find the vertical, horizontal or slant asymptotes for `R(x) = (3x) / (x^2 - 9)`?

Answer 1

vertical asymptotes at x = ± 3
horizontal asymptote at y = 0

Explanation:

Vertical asymptotes occur when the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.

solve : `x^2 - 9 = 0 → (x-3)(x+3 ) = 0 → x = ± 3`

Horizontal asymptotes occur as `lim_(x→±∞) f(x) → 0`

If the degree of the numerator is less than the degree of the denominator , as in this question, numerator degree 1 , denominator degree 2. Then the equation is always y = 0.

Here is the graph of the function to illustrate.
graph{3x/(x^2-9) [-10, 10, -5, 5]}