Question

How do you find vertical, horizontal and oblique asymptotes for `( x^2 - 1)/(x)`?

Answer 1

vertical asymptote x = 0
oblique asymptote y = x

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

`rArr x = 0 " is the asymptote "`

Horizontal asymptotes occur if the degree of the numerator ≤ degree of the denominator. This is not the case here hence there are no horizontal asymptotes.

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is the case here hence there is an oblique asymptote.

To find equation divide numerator by denominator.

`rArr (x^2 - 1)/x = x^2/x - 1/x = x - 1/x`

as `x to+-oo , 1/x to 0" and " y to x`

`rArr y = x " is the asymptote "`
graph{(x^2-1)/x [-10, 10, -5, 5]}