Question

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

Answer 1

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]}