Question 152873
What is the equation of the slant asymptote for {{{R(x) = (x^2-6x+8)/(x-4)}}}?
a. {{{R(x) = x – 2}}}
b. {{{R(x) = 1}}}
c. {{{R(x) = -5x + 2}}}
d. There is no slant asymptote
<pre><font size = 4 color = "indigo"><b>
A rational function has a slant asymptote if and only if it meets
both these conditions:

1. The degree of the the numerator is 1 more than the degree of
the denominator.

2. When the denominator is divided into the numerator, a non-zero
remainder is obtained.  

3. If so then {{{y = quotient}}} is the equation of
the slant asymptote.

--------------------------------------

{{{R(x) = (x^2-6x+8)/(x-4)}}}?

The degree of the numerator is 2 (largest power of x is 2)
The degree of the denominator is 1 (largest power of x is 1)

Therefore it meets the first condition. Let's see if it meets
the second condition:

     <u>       x - 2</u>
x - 4)x² - 6x + 8
      <u>x² - 4x</u>
          -2x + 8
          <u>-2x + 8</u>
                0

No, it leaves a zero remainder.  So R(x) does not have a
slant asymptote.

The correct choice is d
Edwin</pre>