What are the asymptotes and removable discontinuities, if any, of #f(x)= (3x)^2/(x^2-x-6)+3 #?

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Answer 1 · 2017-11-22T12:37:21

vertical asymptotes are #x=3# and #x=-2#
horizontal asymptote is #y=12#

none removable discontinuities ("holes")

#f_((x))={(3x)^2}/{x^2-x-6}+3=#

#= {9x^2}/{(x-3)(x+2)}+3=#

#= {9x^2+3x^2-3x-18}/{(x-3)(x+2)}=#

#= {12x^2-3x-18}/{(x-3)(x+2)}=#

#= 3{4x^2-x-6}/{(x-3)(x+2)}#

#x_(u_(1,2))={3+-sqrt(9+4*12*18)}/24={3+-3sqrt(97)}/24={1+-sqrt(97)}/8#

#x_(u_(1,2))~~1.356 , -1.106 != 3,-2#

#=>#

vertical asymptotes are #x=3# and #x=-2#

#lim_{x rarr +-oo}f_((x))=12#

#=>#

horizontal asymptote is #y=12#