How do you find vertical, horizontal and oblique asymptotes for y=12-(6x)/(1-2x)?

Nov 20, 2016

The vertical asymptote is $x = \frac{1}{2}$
No oblique asymptote
A horizontal asymptote is $y = 15$

Explanation:

The domain of y, is ${D}_{y} = \mathbb{R} - \left\{\frac{1}{2}\right\}$

As you cannot divide by $0$, $x \ne \frac{1}{2}$

Therefore, $x = \frac{1}{2}$ is a vertical asymptote

Let's rewrite the expression

$y = 12 - \frac{6 x}{1 - 2 x} = \frac{12 \left(1 - 2 x\right) - 6 x}{1 - 2 x}$

$= \frac{12 - 24 x - 6 x}{1 - 2 x} = \frac{12 - 30 x}{1 - 2 x}$

As the degree of the numerator $=$ to the degree of the denominator, there is no oblique asymptote.

For calculating the limits, we take the term of highest degree

${\lim}_{x \to \pm \infty} y = {\lim}_{x \to \pm \infty} \frac{- 30 x}{- 2 x} = 15$

A horizontal asymptote is $y = 15$

graph{(y-(12-30x)/(1-2x))(y-15)=0 [-12.87, 15.61, 6.47, 20.71]}