Question

How do you find all the 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 , let the denominator equal zero.

solve : x + 4 = 0 → x = -4 is the asymptote

Horizontal asymptotes occur as `lim_(xto+-oo) f(x) → 0`

divide terms on numerator/denominator by x

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

as `x tooo , 4/x to 0`

`rArr y = 3/1 = 3 " is the asymptote "`

Here is the graph of the function.
graph{(3x)/(x+4) [-20, 20, -10, 10]}