INTRODUCTORY EXERCISES

(1) Find all asymptotes for the given functions.

           2
 (a) y = -----
         x - 4

           x
 (b) y = -------
         3x + 2

         (x - 1)(x - 2)
 (c) y = --------------
         (x - 3)(x - 4)

           2x
 (d) y = ------
         x4 + 1

 (a)       2
     y = -----
         x - 4

     must find out where x - 4 = 0  ==> x = 4

     There is a vertical asymtote at x = 4

           2(1/x)          2/x
     y = ----------  =  ---------  ==> there is a horizontal asymtote at y = 0
         (x-4)(1/x)     1 - (4/x)

     The asymtotes are at x = 4 and at y = 0

 (b)       x
     y = ------  
         3x + 2

     3x + 2 = 0  ==>  x = -2/3  (vertical asymtote)

           x/x          1
     y = -------  =  -------  ==> y = 1/3 is a horizontal asymtote.
         3 + 2/x     3 + 2/x

     There is a vertical asymtote at x = -2/3 and a horizontal asymtote at y = 1/3.


 (c)     (x-1)(x-2)
     y = ----------
         (x-3)(x-4)

     (x-3)(x-4) = 0  ==>  x = 3 or x = 4  (These are vertical asymtotes)

         (1 - 1/x)(1 - 2/x)
     y = ------------------  ==>  y = 1 (horizontal asymtote)
         (1 - 3/x)(1 - 4/x)

     The asymtotes are at x = 3, x = 4 and y = 1


 (d)       2x
     y = ------
         x4 + 1

     x4 + 1 = 0  ==> x4 = -1  impossible.  Therefore no vertical asymtotes.

            2x/x4
     y = ------------
         x4/x4 + 1/x4

           2/x3
       = --------   ==>  y = 0 is a horizontal asymtote.
         1 + 1/x4

     So there is one asymtote at y = 0.