Mathematics

Problem

Consider the function

r(x)=\dfrac{x^{2}-4}{x(x-2)}

. Which statement correctly describes the vertical asymptotes and one-sided behavior of

r

? a)

r

has vertical asymptotes at

x=0

and

x=2

; at each asymptote

r(x)\to\infty

from the left and

r(x)\to-\infty

from the right. b)

r

has a vertical asymptote at

x=0

with

\displaystyle\lim_{x\to 0^-}r(x)=-\infty

and

\displaystyle\lim_{x\to 0^+}r(x)=\infty

, and a removable discontinuity at

x=2

. c)

r

has a vertical asymptote at

x=0

only, where

r(x)\to\infty

from both sides. d)

r

has no vertical asymptotes because the numerator and denominator share a common factor.