Question

How do you find vertical, horizontal and oblique asymptotes for `(3x )/( 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

`f(x)=((3x)/x)/(x/x+4/x)=3/(1+4/x)`

as `xto+-oo,f(x)to3/(1+0)`

`rArry=3" is the asymptote"`

Oblique 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 oblique asymptotes.
graph{(3x)/(x+4) [-20, 20, -10, 10]}