Question

How do you find the asymptotes for `(2x^2 - x - 38) / (x^2 - 4)`?

Answer 1

vertical asymptotes at x = ± 2

horizontal asymptote at y = 2

Explanation:

As the denominator of a rational function tends to 0 there
will be a vertical asymptote.

solve `(x^2 - 4) =0`

`(x-2) = 0 or (x+2) = 0` hence vertical asymptotes at `x = ± 2`

[horizontal asymptotes occur when `lim_(x→±∞) f(x) → 0`]

when the degree of the numerator and denominator are equal

the equation can be found by taking the ratio of
leading coefficients.

In this question they are equal , both of degree 2

and so `y = 2/1 rArr y = 2`

The graph shows the asymptotes.
graph{(2x^2-x-38)/(x^2-4) [-10, 10, -5, 5]}