Question

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

Answer 1

vertical asymptotes at x = ± 2
horizontal asymptote at y = 0

Explanation:

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

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

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

If the degree of the numerator is less than the degree of the denominator, as in this case (degree 1 < degree 2 ) then the equation is y = 0.

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