How do you find the vertical, horizontal or slant asymptotes for #f(x) = ((x-3)(9x+4))/(x^2-4)#?

1 Answer
Nov 8, 2016

Answer:

The vertical asymptotes are #x=2# and #x=-2#
The horizontal asymptote is #y=9#

Explanation:

The denominator, #x^2-4=(x+2)(x-2)#
As we cannot divide by zero, the vertical asymptotes are #x=2# and #x=-2#
As the degree of the numerator is the same asthe degree of the denominator, there is no slant asymptote:
#lim_(x->+-oo)f(x)=lim_(x->+-oo)(9x^2)/x^2=9#

#:. y=9# is a horizontal asymptote
graph{(y-((x-3)(9x+4))/((x+2)(x-2)))(y-9)=0 [-41.13, 41.15, -20.55, 20.57]}