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

Question

Serenity Johnson · Accepted Answer

The vertical asymptote is #x=1/2#
No oblique asymptote
A horizontal asymptote is #y=15# 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]}

Chloe Alexander · Answer

undefined To find the vertical, horizontal, and oblique asymptotes for the function $y = \frac{{12 - 6x}}{{1 - 2x}}$:

1. Vertical Asymptotes:
   Vertical asymptotes occur where the denominator of the rational function becomes zero, but the numerator does not. Set the denominator equal to zero and solve for $x$. In this case, set $1 - 2x = 0$ and solve for $x$. The solution is the vertical asymptote.

2. Horizontal Asymptote:
   To find the horizontal asymptote, look at the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis (y = 0). If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients to find the horizontal asymptote. If the degree of the numerator is greater, there is no horizontal asymptote.

3. Oblique Asymptote:
   If the degree of the numerator is one greater than the degree of the denominator, there is an oblique asymptote. To find it, perform polynomial long division to divide the numerator by the denominator. The quotient represents the oblique asymptote.

Once you find the vertical, horizontal, and oblique asymptotes, you have a good understanding of the behavior of the function as $x$ approaches infinity or negative infinity.