# How do you find the asymptotes for 1. f(x) = (3x) / (x+4)?

Mar 6, 2016

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.

$\Rightarrow y = \frac{3}{1} = 3 \Rightarrow y = 3 \text{ is the equation }$

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

Mar 6, 2016

Just another way of looking at the same thing

#### Explanation:

Given:$\text{ } f \left(x\right) = \frac{3 x}{x + 4}$

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

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 $x \to \infty$ then $\frac{4}{x} \to 0$ so we end up with

${\lim}_{x \to \pm \infty} f \left(x\right) = {\lim}_{x \to \pm \infty} \frac{3}{1 + \frac{4}{x}} = + 3$