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

By actual division,

This form reveals ths the asymptotes as follows.

y = quotient = x + 2 gives the slant asymptote.

The denominator in the remainder,

There is no horizontal asymptote.

A reorganization of the equation gives the form

with the pair of asymptotes

( y-x-2)(x-2)= 0

graph{y(x-2)-x^2=0 [-80, 80, -40, 40]}

To find the vertical asymptote(s) of a rational function, set the denominator equal to zero and solve for x. In this case, the denominator is x - 2. So, set x - 2 = 0 and solve for x. The solution is x = 2. Therefore, the vertical asymptote is x = 2.

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients to find the horizontal asymptote. In this case, the degrees of the numerator and the denominator are both 1. So, divide the leading coefficient of the numerator by the leading coefficient of the denominator, which is (1/1). Therefore, the horizontal asymptote is y = x.

To find the slant asymptote, perform polynomial long division on the function. Divide the numerator by the denominator. In this case, divide x^2 by (x - 2). The quotient will give the equation of the slant asymptote. Therefore, the slant asymptote is y = x + 2.