# How do you find vertical, horizontal and oblique asymptotes for #( 4x^5)/(x^3-1)#?

##### 1 Answer

Aug 8, 2018

This function has a vertical asymptote

#### Explanation:

Given:

#f(x) = (4x^5)/(x^3-1)#

Note that:

#(4x^5)/(x^3-1)=(4x^5-4x^2+4x^2)/(x^3-1)#

#color(white)((4x^5)/(x^3-1))=(4x^2(x^3-1)+4x^2)/(x^3-1)#

#color(white)((4x^5)/(x^3-1))=4x^2+(4x^2)/(x^3-1)#

#color(white)((4x^5)/(x^3-1))=4x^2+(4x^2)/((x-1)(x^2+x+1))#

The only real zero of the denominator is

Hence,

Note also that:

#lim_(x->+-oo) (4x^2)/(x^3-1) = lim_(x->+-oo) (4/x)/(1-1/x^3) = 0/(1-0) = 0#

Hence

graph{(4x^5)/(x^3-1) [-5, 5, -35, 75]}