Question

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

Answer 1

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]}