Question

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

Answer 1

The vertical asymptote is `x=1/2`
No oblique asymptote
A horizontal asymptote is `y=15`

Explanation:

The domain of y, is `D_y=RR-{1/2}`

As you cannot divide by `0`, `x!=1/2`

Therefore, `x=1/2` is a vertical asymptote

Let's rewrite the expression

`y=12-(6x)/(1-2x)=(12(1-2x)-6x)/(1-2x)`

`=(12-24x-6x)/(1-2x)=(12-30x)/(1-2x)`

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->+-oo)y=lim_(x->+-oo)(-30x)/(-2x)=15`

A horizontal asymptote is `y=15`

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