# How do you find vertical, horizontal and oblique asymptotes for (3x-12)/(4x-2)?

May 6, 2016

vertical asymptote $x = \frac{1}{2}$
horizontal asymptote $y = \frac{3}{4}$

#### Explanation:

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

solve : 4x - 2 = 0 → 4x = 2

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

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

divide terms on numerator/denominator by x

$\frac{\frac{3 x}{x} - \frac{12}{x}}{\frac{4 x}{x} - \frac{2}{x}} = \frac{3 - \frac{12}{x}}{4 - \frac{2}{x}}$

as $x \to \pm \infty , y \to \frac{3 - 0}{4 - 0}$

$\Rightarrow y = \frac{3}{4} \text{ is the asymptote }$

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 1 and denominator-degree 1)hence there are no oblique asymptotes.
graph{(3x-12)/(4x-2) [-10, 10, -5, 5]}