# How do you find the Vertical, Horizontal, and Oblique Asymptote given g(x)=( x+3) / (x(x-5))?

Jul 19, 2016

vertical asymptotes x = 0 , x = 5
horizontal asymptote y = 0

#### Explanation:

The denominator of g(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

solve: x(x -5) = 0 → x = 0 , x = 5 are the asymptotes

Horizontal asymptotes occur as

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

$g \left(x\right) = \frac{x + 3}{{x}^{2} - 5 x}$

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

$\frac{\frac{x}{x} ^ 2 + \frac{3}{x} ^ 2}{{x}^{2} / {x}^{2} - \frac{5 x}{x} ^ 2} = \frac{\frac{1}{x} + \frac{3}{x} ^ 2}{1 - \frac{5}{x}}$

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

$\Rightarrow y = 0 \text{ is the asymptote}$

Oblique asymptotes occur when the degree of the numerator > degree of denominator. This is not the case here (numerator-degree 1 , denominator-degree 2) Hence there are no oblique asymptotes.
graph{(x+3)/(x(x-5)) [-10, 10, -5, 5]}