Question

How do you find the asymptotes for `f(x) = (3x )/ (x+4)`?

Answer 1

vertical asymptote at x = -4
horizontal asymptote at y = 3

Explanation:

vertical asymptotes occur as the denominator of a rational function tends to zero.

To find it's equation let the denominator equal zero.

solve : x + 4 = 0 hence x = -4 is the equation of asymptote.

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

If the degree of the numerator and denominator are equal the equation of the asymptote can be found by taking the ratio of leading coefficients.
In this function they are both of degree 1.

hence equation is `y = 3/1 = 3`
Here is the graph of f(x) as an illustration.
graph{3x/(x+4) [-20, 20, -10, 10]}