How do you find the vertical, horizontal or slant asymptotes for #g(x) = [(3x-1)(x-5)^2] / [x(4x-1)(2x-7)]#?

1 Answer
Nov 8, 2016

Answer:

There are 3 vertical asymptotes #x=0 ; x=1/4 ; and x=7/2#
There is a horizontal asymptote #y=3/8#

Explanation:

As we cannot divide by (0), there are 3 vertical asymptotes,
#x=0#, and #x=1/4# and #x=7/2#
As the degree of the numerator #=# the degree of the denominator, there is no slant asymptotes.
#lim_(x->oo)g(x)=lim_(x->oo)(3x^3)/(8x^3)=3/8#
So, there is a horizontal asymptote #y=3/8#
graph{(3x-1)(x-5)^2/((x)(4x-1)(2x-7)) [-10.09, 15.22, -6.06, 6.6]}