How do you find the vertical, horizontal or slant asymptotes for $f (x) = \frac{x^{2}}{x - 2}$ $f(x) = (x^2) / (x-2)$ ?

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Answer 1 · 2016-12-17T06:50:33

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

By actual division,

$y = f (x) = x + 2 + \frac{4}{x - 2}$ $y = f(x) = x+2+4/(x-2)$

This form reveals ths the asymptotes as follows.

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

The denominator in the remainder,

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

There is no horizontal asymptote.

A reorganization of the equation gives the form

$(y - x - 2) (x - 2) = c o n s tan t = 4$ $(y-x-2)(x-2)=constant = 4$ 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]}

How do you find the vertical, horizontal or slant asymptotes for f(x) = (x^2) / (x-2)f(x)=x2x−2f(x) = (x^2) / (x-2)?