A value of #x# is a discontinuity for a rational function if it makes both the numerator and the denominator equal to 0. For this rational function, both the numerator and denominator have factors of #(x+2)#, meaning that when #x=–2# both sides of the fraction will equal 0. So the function has a discontinuity at #x=–2.#
Any value of #x# that makes just the denominator equal to 0 marks the location of an asymptote for the rational function. Here, the denominator has factors of #(x-4)# and #(x+4)# which do not appear in the numerator. So when #x=–4# or #x=4,# the denominator will equal 0. Thus, these values of #x# mark asymptotes for the rational function. (The equations of the asymptotes are simply #x=–4# and #x=4.)#
