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

Nov 8, 2016

There are 3 vertical asymptotes x=0 ; x=1/4 ; and x=7/2
There is a horizontal asymptote $y = \frac{3}{8}$

#### Explanation:

As we cannot divide by (0), there are 3 vertical asymptotes,
$x = 0$, and $x = \frac{1}{4}$ and $x = \frac{7}{2}$
As the degree of the numerator $=$ the degree of the denominator, there is no slant asymptotes.
${\lim}_{x \to \infty} g \left(x\right) = {\lim}_{x \to \infty} \frac{3 {x}^{3}}{8 {x}^{3}} = \frac{3}{8}$
So, there is a horizontal asymptote $y = \frac{3}{8}$
graph{(3x-1)(x-5)^2/((x)(4x-1)(2x-7)) [-10.09, 15.22, -6.06, 6.6]}
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