# How do you find the horizontal asymptote for g(x)=(x+3)/(x(x+4))?

Mar 22, 2018

$y = 0$

#### Explanation:

$\text{horizontal asymptotes occur as}$

${\lim}_{x \to \pm \infty} g \left(x\right) \to c \text{ (a constant)}$

$\text{divide terms on numerator/denominator by the highest}$
$\text{power of x, that is } {x}^{2}$

$g \left(x\right) = \frac{\frac{x}{x} ^ 2 + \frac{3}{x} ^ 2}{{x}^{2} / {x}^{2} + \frac{4 x}{x} ^ 2} = \frac{\frac{1}{x} + \frac{3}{x} ^ 2}{1 + \frac{4}{x}}$

$\text{as } x \to \pm \infty , g \left(x\right) \to \frac{0 + 0}{1 + 0}$

$\Rightarrow y = 0 \text{ is the asymptote}$
graph{(x+3)/(x^2+4x) [-10, 10, -5, 5]}