How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)= (x^2)/(x-2)^2#?

1 Answer
Aug 3, 2017

Answer:

The vertical asymptote is #x=2#
The horizontal asymptote is #y=1#
No oblique asymptote

Explanation:

The vertical asymptotes are calculated by performing the limits

#lim_(x->-2^(-))f(x)=lim_(x->-2^(-))x^2/(x-2)^2= 4/(0^+) = +oo#

#lim_(x->-2^(+))f(x)=lim_(x->-2^(+))x^2/(x-2)^2= 4/(0^+) = +oo#

The vertical asymptote is #x=2#

To determine the horizontal asymptote, we calculate

#lim_(x->-oo)f(x)=lim_(x->-oo)x^2/(x^2)=1#

The horizontal asymptote is #y=1#

As the degree of the numerator is #=# to the degree of the denominator, there is no oblique asymptote

graph{x^2/(x-2)^2 [-20.27, 20.28, -10.14, 10.14]}