# How do you find the vertical, horizontal or slant asymptotes for (x-5)/(x+1)?

Mar 15, 2016

vertical asymptote x = - 1
horizontal asymptote y = 1

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find equation let denominator equal zero.

solve : x + 1 = 0 → x = -1 is the asymptote

Horizontal asymptotes occur as lim_(x→±∞) f(x) → 0

divide all terms on numerator / denominator by x

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

now as x → ∞ ,  5/x" and " 1/x → 0

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

Here is the graph of the function
graph{(x-5)/(x+1) [-20, 20, -10, 10]}