# How do you find the asymptotes for y= (x + 1 )/ (2x - 4)?

Feb 3, 2016

There is a vertical asymptote at $x = 2$.
There is a horizontal asymptote at $y = \frac{1}{2}$.

#### Explanation:

First step is always to simplify what you have.

$\frac{x + 1}{2 x - 4} \equiv \frac{1}{2} \left(1 + \frac{3}{x - 2}\right)$

As you know, dividing by a very large number results in almost zero. Therefore,

${\lim}_{x \to - \infty} \frac{x + 1}{2 x - 4} = {\lim}_{x \to - \infty} \frac{1}{2} \left(1 + \frac{3}{x - 2}\right)$
$= \frac{1}{2} \left(1 + 0\right) = \frac{1}{2}$

Similarly,

${\lim}_{x \to \infty} \frac{x + 1}{2 x - 4} = \frac{1}{2}$.

Hence there is a horizontal asymptote at $y = \frac{1}{2}$.

Also note that $x \ne 2$, as that will result in division by zero.

There is a vertical asymptote at $x = 2$.

Please refer to the graph below.

graph{(x+1)/(2x-4) [-10, 10, -5, 5]}