ADVANCED EXERCISES

(1) Find all asymptotes and sketch the function.

                x3 + 5
  (a) f(x) = ------------
              x2 + 3x + 1

                x2
  (b) g(x) = -------
              x - 3 

           x3 - 4x2 - 49x - 90
  (c) y = ---------------------
              2x2 +12x + 18

              4x5 - 6
  (d) h(x) = ----------
              9x5 + 7x2

             x4 - 3x3 + 5x2 - 7x + 9
  (e) y = -----------------------------
           x5 - x4 - x3 + 3x2 - 5x + 18

SOLUTION:

                x3 + 5
  (a) f(x) = ------------
              x2 + 3x + 1

    x2 + 3x + 1 = 0
         -3  
    x = ---------  (2 vertical asymptotes)
            2

             (x3/x3) + (5/x3) 
    y = ------------------------- = undefined (no horizontal asymptotes)
        (x2/x3) + (3x/x3) + (1/x3) 
 
	           x - 3 + ((8x + 8)/(x2 + 3x + 1))     
                 -------------------
    x2 + 3x + 1 / x3 + 0x2 + 0x + 5	
                  x3 + 3x2 + x
                  ------------
                      -3x2 - x + 5
                      -3x2 -9x - 3
                      ------------
                            8x + 8  

                    8x/x2  + 8/x2 
    y = x - 3 + -------------------- 
                x2/x2  + 3x/x2 + 1/x2   
      = x - 3 + 0 
      = x - 3 (one oblique asymptote)




                x2
  (b) g(x) = -------
              x - 3 

   x - 3 = 0
   x = 3 (one vertical asymptote)

           x2/x2
   y = ------------ = undefined (no horizontal asymptotes)
        x/x2 - 3/x2 

           x + ((3x)/(x - 3))     
          -------------------
   x - 3 / x2 + 0x + 0	
           x2 - 3x
           ------------
                3x 

             3x/x
   y = x + --------- = x + 3 (one oblique asymptote)
           x/x - 3/x




           x3 - 4x2 - 49x - 90
  (c) y = ---------------------
              2x2 +12x + 18

   2x2 +12x + 18 = 2(x2 + 6x + 9) = 0 
   x = -3 (one vertical asymptote)

        x3/x3 - 4x2/x3 - 49x/x3 - 90/x3
   y = -------------------------------- = undefined (no horizontal asymptotes)
            2x2/x3 +12x/x3 + 18/x3 

                    0.5x - 5 + ((2x)/(2x2 + 12x + 18))     
                   -----------------------------------
   2x2 + 12x + 18 / x3 - 4x2 - 49x - 90	
                    x3 + 6x2 + 9x
                    --------------
                      -10x2 - 58x - 90
                      -10x2 - 60x - 90
                      ----------------
                               2x  

                            2x/2
   y = 0.5x - 5 + -----------------------  
                   2x2/x2 +12x/x2 + 18/x2 
      = 0.5x - 5 + 0 
      = 0.5x - 5 (one oblique asymptote)
   



              4x5 - 6
  (d) h(x) = ----------
              9x5 + 7x2

   9x5 + 7x2 = x2(9x3 + 7) = 0
   x = (-7/9)1/3 or x = 0 (two vertical asymptotes)

         4x5/x5 - 6/x5 
   y = ---------------- = 4/9 (one horizontal asymptote)
        9x5/x5 + 7x2/x5 

  There are no oblique asymptotes, as the degree of the numerator 
  is not one greater than the degree of the denominator




             x4 - 3x3 + 5x2 - 7x + 9
  (e) y = -----------------------------
           x5 - x4 - x3 + 3x2 - 5x + 18

   First, reduce the equation to y = 1/(x + 2)

   x + 2 = 0
   x = -2 (one vertical asymptote)

          1/x   
   y = --------- = 0 (one horizontal asymptote)
       x/x + 2/x

  There are no oblique asymptotes, as the degree of the numerator 
  is not one greater than the degree of the denominator