MODERATE EXERCISES

(1) Find all asymptotes for the given functions.

            2x2 + 3x
 (a) g(x) = ---------
            3x2 - 48

                5x4
 (b) R(x) = ------------
            2x2 + 3x - 2

                 3x
 (c) F(x) = -------------
            x4 + 2x2 + 1


           x2 + 2
 (d) H(x)= ------
           x - 1

            2x2 + 3x
 (a) g(x) = ---------
            3x2 - 48

     3x2 - 48 = 0
     x2 = 16
     x = 4(vertical asymtotes)

            2x2/x2 + 3x/x2
     g(x) = --------------
            3x2/x2 - 48/x2

             2 + 3/x
          = ---------
            3 - 48/x2

          y= 2/3 (horizontal asymtote)

     The asymtotes of g(x) are x = 4, x = -4 and y = 2/3 


 (b)            5x4
     R(x) = ------------
            2x2 + 3x - 2

     2x2 + 3x - 2 = 0
     (x + 2)(2x - 1) = 0
     x = -2 or x = 1/2  (vertical asymtotes)

                   5x4/x4
     R(x) = --------------------
            2x2/x4 + 3x/x4 - 2/x4

                     5
          = ------------------
            2/x2 + 3/x3 - 2/x4

          ==> no horizontal asymtotes

     There are two vertical asymtotes, one at x = -2 and the other at x = 1/2.


 (c)             3x
     F(x) = ------------
            x4 + 2x2 + 1

     x4 + 2x2 + 1 = 0
     (x2 + 1)2 = 0
     x2 = -1  impossible.  Therefore no vertical asymtotes

                3x/x4
     F(x) = --------------
            x4/x4 + 2x2/x4 + 1/x4

                3/x3
          = -----------
            1 + 2/x2 + 1/x4

          y = 0 is a horizontal asymtote.

     There is one asymtote which is at y = 0.

 (d)
          x2 + 2
    H(x)= ------
          x - 1

     x - 1 = 0  ==>  x = 1 is a vertical asymtote.

         x2/x2 + 2/x2      1 + 2/x2
    H(x)= ------------  =  ----------
         x/x2 - 1/x2       1/x - 1/x2

     ==> no horizontal asymptotes.

     There is only one asymtote which is a x = 1.