Question

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

Answer 1

Slant: y = x + 2.
Vertical: x = 2.

Explanation:

By actual division,

`y = f(x) = x+2+4/(x-2)`

This form reveals ths the asymptotes as follows.

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

The denominator in the remainder,

`x-2=0` gives the vertical asymptote.

There is no horizontal asymptote.

A reorganization of the equation gives the form

`(y-x-2)(x-2)=constant = 4` that represents a hyperbola,

with the pair of asymptotes

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

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