Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = (3x - 12 ) / ( x + 4)`?

Answer 1

vertical asymptote x = -4
horizontal asymptote y = 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 + 4 = 0 → x = -4 is the asymptote

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" (a constant)"`

divide terms on numerator/denominator by x

`((3x)/x-12/x)/(x/x+4/x)=(3-12/x)/(1+4/x)`

as `xto+-oo,f(x)to(3-0)/(1+0)`

`rArry = 3/1=3" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1). Hence there are no slant asymptotes.
graph{(3x-12)/(x+4) [-20, 20, -10, 10]}