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:

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

solve `x^2 -4 = 0 rArr (x-2)(x+2) = 0 rArr x = ± 2`

Horizontal asymptotes occur as `lim_(x→±∞) f(x) → 0`

If the degree of the numerator and denominator are equal, then the equation can be found by taking the ratio of leading coefficients.
Here they are both of degree 2.

`rArr y = 2/1 =2`
Here is the graph of the function as an illustration.
graph{(2x^2-x-38)/(x^2-4) [-10, 10, -5, 5]}