Name__________________________ 
                          AP Calculus BC                    Summer Review Packet (Limits & Derivatives) 
                 Limits 
                 1.       Answer the following questions using the graph of ƒ(x) given below. 
                  
                  
                  
                  
                  
                  
                  
                  
                  
                  
                  
                  
                  
                  
                  
                  
                          (a)     Find ƒ(0)                                  (b)     Find ƒ(3) 
                  
                  
                  
                  
                  
                          (c)     Find                                       (d)     Find              
                                         lim fx( )                                         lim fx( )
                                        x5                                               x0
                  
                  
                  
                  
                  
                          (e)     Find                                       (f)     Find               
                                         lim fx( )                                          lim fx( )
                                                                                               
                                        x3                                                x3
                  
                  
                  
                  
                  
                          (g)     List all x-values for which ƒ(x) has a removable discontinuity.  Explain what  
                                  section(s) of the definition of continuity is (are) violated at these points. 
               
               
               
               
                     (h)   List all x-values for which ƒ(x) has a nonremovable discontinuity.  Explain what  
                           section(s) of the definition of continuity is (are) violated at these points. 
                      
               
               
               
               
              In problems 2-10, find the limit (if it exists) using analytic methods (i.e. without using a 
              calculator). 
                           2                                            1cos2 x
                         3xx21   30
              2.     lim      3                              3.     lim                        
                     x2    x 8                                   x/6  4x
               
               
               
               
               
               
                                                                        1/(x1)  1
                     lim  x31                                                
              4.                                             5.     lim                                             
                     x4   x4                                      x0      x
                                                                                                                                                       3
                                                       1/ 1x                1                                                               sin6
                                                     
                              6.             lim                                                                       7.             lim                    
                                              x0                 x                                                                   0         7
                               
                               
                               
                               
                               
                               
                                                           22                                                                                  6x36
                              8.             limsin 3t                                                                 9.              lim                       
                                              t0         t3                                                                          x6        6x
                                              
                               
                               
                               
                               
                               
                              10.             lim sin(( /6)x)(1/2)  
                                              x   0                     x
                                             Hint:          sin( )sincos cossin                                                                                                         
                                                 2x1        3,0x 
                  11.      Suppose  fx()                               . 
                                                      x1
                                                        2
                                                    4x k,x      0
                                                
                           (a)      For what value of k will f be piecewise continuous at x = 0?  Explain why this is  
                                    true using one-sided limits.  (Hint:  A function is continuous at  
                                    x =c if (1) f(c) exists, (2)            exists, and (3)                       .) 
                                                                  lim fx( )                    lim f (x)  f (c)
                                                                  xc                          xc
                   
                   
                   
                   
                            
                           (b)      Using the value of k that you found in part (a), accurately graph f below.    
                                    Approximate the value of                    
                                                                    lim fx( )
                                                                     x1
                                     lim fx( )  _______________ 
                                     x1
                                                                                  
                                                                                        
                            
                           (c)      Rationalize the numerator to find                    analytically. 
                                                                              lim fx( )
                                                                              x1