# How do you find the vertical, horizontal or slant asymptotes for f(x)= (x+5)/(x+3)?

Jul 19, 2016

vertical asymptote x = -3
horizontal asymptote y = 1

#### Explanation:

The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: x + 3 = 0 → x = -3 is the asymptote

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$\frac{\frac{x}{x} + \frac{5}{x}}{\frac{x}{x} + \frac{3}{x}} = \frac{1 + \frac{5}{x}}{1 + \frac{3}{x}}$

as $x \to \pm 00 , f \left(x\right) \to \frac{1 + 0}{1 + 0}$

$\Rightarrow y = 1 \text{ is the asymptote}$

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