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

Mar 30, 2016

vertical asymptote x = - 3
horizontal asymptote y = 1

#### 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 + 3 = 0 → x = -3 is the asymptote

Horizontal asymptotes occur as ${\lim}_{x \to \infty} f \left(x\right) \to 0$

divide terms on numerator/denominator by x

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

As x$\to \infty , \frac{5}{x} \text{ and } \frac{3}{x} \to 0$

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

Here is the graph of f(x).
graph{(x+5)/(x+3) [-10, 10, -5, 5]}