Question

How do you find the asymptotes for `1. f(x) = (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 equation

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

If the degree of the numerator and denominator are equal , as is the case here , both of degree 1. The equation can be found by taking the ratio of the leading coefficients.

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

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

Answer 2

Just another way of looking at the same thing

Explanation:

Given:`" "f(x)=(3x)/(x+4)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Lets 'get rid' if the `x` in the numerator!

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

Straight away you can observe that the denominator is 0 when `x=-4`. Thus the expression is undefined. The result is an asymptote at `x=-4`

As absolute `x ->oo` then `4/x->0` so we end up with

`lim_(x->+-oo) f(x) = lim_(x->+-oo) 3/(1+4/x) =+3`