Question

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

Answer 1

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`