What are the asymptotes and removable discontinuities, if any, of #f(x)=((3x^2 -1)/(2x^2 -5x + 3))#?

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Answer 1 · 2017-11-21T16:11:54

vertical asymptotes are #x=1# and #x=1 1/2#

horizontal asymptote is #y=1 1/2#

no removable discontinuities ("holes")

#f_((x))=(3x^2-1)/(2x^2-5x+3)=(3x^2-1)/((2x-3)(x-1))#

#x_(d_1)=3/2#

#x_(d_2)=1#

#x_u=+-1/sqrt3#

#=>#
#x_(d_1)!=x_(d_2)!=x_u#
#=>#
there're no holes

#=>#
vertical asymptotes are #x=1# and #x=1 1/2#

#lim_(x rarr +-oo)f_((x))=1 1/2#

#=>#
horizontal asymptote is #y=1 1/2#

graph{(3x^2-1)/(2x^2-5x+3) [-17.42, 18.62, -2.19, 15.83]}