How do you find the Vertical, Horizontal, and Oblique Asymptote given $f (x) = \frac{x^{2}}{{(x - 2)}^{2}}$ $f(x)= (x^2)/(x-2)^2$ ?

Question

1441 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-08-03T09:33:17

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

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]}

How do you find the Vertical, Horizontal, and Oblique Asymptote given f(x)= (x^2)/(x-2)^2f(x)=x2(x−2)2f(x)= (x^2)/(x-2)^2?