What are the asymptotes of g(x) = (x-1)/(x-x^3)?

1 Answer
Jun 16, 2018

The vertical asymptotes are -1, 0 and 1, The horizontal asymptote is y=0.

Explanation:

Vertical asymptotes:
x-x^3 =0 then x(1-x^2)=0 and x(1-x)(1+x)=0
Are: x_1=0; x_2=1 and x_3 =-1

Horizontal asymptote:
Lim_{x \rightarrow + \infty} \frac{x-1}{x-x^3} = \frac{+ \infty}{- \infty} IND
Lim_{x \rightarrow + \infty} \frac{x-1}{x(1-x)(1+x)} = Lim_{x \rightarrow + \infty} \frac{1}{x(1+x)} = \frac{1}{+ \infty} = 0

Lim_{x \rightarrow -\infty} \frac{x-1}{x-x^3} = \frac{-\infty}{ \infty} IND
Lim_{x \rightarrow - \infty} \frac{x-1}{x(1-x)(1+x)} = Lim_{x \rightarrow - \infty} \frac{1}{x(1+x)} = \frac{1}{- \infty} = 0