How do you find the vertical, horizontal or slant asymptotes for #f(x) = (8x-12)/( 4x-2)#?

Question

Liam Brennan · Accepted Answer

There is a horizontal asymptote #y = 2#.

There is a vertical asymptote #x = 1/2#.

This is the graph of #f(x)#. graph{(4x-2) [-20, 20, -10, 10]} / (8x-12) First rewrite #f(x)# by simplifying it. #f(x) = frac{8x-12}{4x-2}# #= 2-frac{4}{2x-1}# It is evident to you that #lim_{x->oo} f(x) = lim_{x->oo} (2-frac{4}{2x-1})# #= 2-0# #= 2# In a similar vein, #lim_{x->-oo} f(x) = 2# There is a horizontal asymptote #y = 2#. #f(x)# is undefined at #x = 1/2#. There is a vertical asymptote #x = 1/2#.

Elijah Allen · Answer

undefined To find the vertical, horizontal, or slant asymptotes for the function $ f(x) = \frac{8x - 12}{4x - 2} $, follow these steps:

1. **Vertical Asymptotes**: Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not simultaneously become zero. Set the denominator equal to zero and solve for $ x $. The vertical asymptote(s) will be the solution(s) obtained.

$$ 4x - 2 = 0 $$
   $$ x = \frac{1}{2} $$

So, the vertical asymptote is $ x = \frac{1}{2} $.

2. **Horizontal Asymptote**: To find the horizontal asymptote, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of both polynomials to find the horizontal asymptote.

In this case, both the numerator and denominator are linear functions. The horizontal asymptote is found by dividing the leading coefficient of the numerator by the leading coefficient of the denominator.

$$ \lim_{x \to \infty} \frac{8x - 12}{4x - 2} = \frac{8}{4} = 2 $$

Therefore, the horizontal asymptote is $ y = 2 $.

3. **Slant Asymptote (Oblique Asymptote)**: A slant asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator. To find the slant asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient obtained will be the equation of the slant asymptote.

Since the degree of the numerator (1) is one greater than the degree of the denominator (1), we proceed with polynomial division:

$$ \frac{8x - 12}{4x - 2} = 2 $$

Therefore, the slant asymptote is the line $ y = 2x $.

To summarize:
- Vertical asymptote: $ x = \frac{1}{2} $
- Horizontal asymptote: $ y = 2 $
- Slant asymptote: $ y = 2x $