# How do you find the Vertical, Horizontal, and Oblique Asymptote given y= (x + 1) / (2x - 4)?

Aug 7, 2016

vertical asymptote x = 2
horizontal asymptote $y = \frac{1}{2}$

#### Explanation:

The denominator of y 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 :$2 x - 4 = 0 \Rightarrow x = 2 \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$\frac{\frac{x}{x} + \frac{1}{x}}{\frac{2 x}{x} - \frac{4}{x}} = \frac{1 + \frac{1}{x}}{2 - \frac{4}{x}}$

as $x \to \pm \infty , y \to \frac{1 + 0}{2 - 0}$

$\Rightarrow y = \frac{1}{2} \text{ is the asymptote}$

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