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]}