Question

How do you find vertical, horizontal and oblique asymptotes for `(x^2 - 5)/( x+3)`?

Answer 1

vertical asymptote x = -3
oblique asymptote y = x - 3

Explanation:

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

solve : x + 3 = 0 `rArr x = -3 " is the asymptote "`

Horizontal asymptotes occur when 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 and to find the equation we require to use 'polynomial division'.

`rArr(x^2-5)/(x+3) = x - 3 + 4/(x+3)`

as `x to +- oo , y to x - 3`

`rArr y = x - 3 " is the asymptote "`
graph{(x^2-5)/(x+3) [-46.24, 46.24, -23.12, 23.1]}