SOLUTION: Given the following rational function: (a) state the domain. (b) find the vertical and horizontal asymptotes, if any. (c) find the oblique asymptotes, if any. (d) submi

f(x) = 

Since the denominator is not a factor of the numerator,
we find all vertical asymptotes by setting the denominator = 0

x - 5 = 0
    x = 5

So the vertical asymptote is the line whose equation is x = 5,
which is a vertical line through 5 on the x-axis, which I will
draw in green:



It has no horizontal asymptote because the numerator has a greater
degree than the denominator.  However since the degree of the
numerator is exactly 1 more than the degree of the denominator,
it does have a oblique (slanted) anymptote.  We find that by dividing
the denominator into the numerator:

            x + 11
x - 5)x² + 6x -  8
      x² - 5x
          11x -  8  
          11x - 55
                47

The oblique asymptote is the line which has the equation

y = x + 11

So we draw that oblique asymptote:



We get a few points and sketch in the graph:



The domain is all real numbers except where there is a vertical asymptote,
which is at 5, so the domain in interval notation is: 

 U 

Edwin