# How do you find vertical, horizontal and oblique asymptotes for y = (x + 1)/(x - 1)?

Mar 23, 2016

vertical asymptote x = 1
horizontal asymptote y = 1

#### Explanation:

Vertical asymptotes occur when the denominator of a rational function tends to zero. To find the equation let the 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 and denominator by x

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

As x →∞ ,  1/x → 0

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

Oblique asymptotes occur when the degree of the numerator is greater than the degree of the denominator. This is not the case here , hence there are no oblique asymptotes.

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