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

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Answer 1 · 2016-12-17T06:50:33

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

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]}