AP Calculus 
                                                                                                             
                                                                                                            	
                                                   Limits,	Continuity,	and	Differentiability	
                                                                                                            	
                                                                                                            	
                                                                                       Student	Handout	
                                                                                                            	
                                                                                                            	
                                                                                                            	
                                                                                                            	
                                                                                                            	
                                                                                         2017‐2018 EDITION 
                                                                                                             
                                                                                                             
                                                                               
                          Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org                                 	
                           
                          Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org                                 	
                                                                  Limits, Continuity, and Differentiability 
                           
                          Students should be able to:  
                                      Determine limits from a graph  
                                      Know the relationship between limits and asymptotes (i.e., limits that become infinite at a 
                                       finite value or finite limits at infinity)  
                                      Compute limits algebraically  
                                      Discuss continuity algebraically and graphically and know its relation to limit. 
                                      Discuss differentiability algebraically and graphically and know its relation to limits and 
                                       continuity  
                                      Recognize the limit definition of derivative and be able to identify the function involved 
                                       and the point at which the derivative is evaluated. For example, since 
                                        f (a)  lim f (a  h) f (a),  recognize that lim cos(  h)cos() is simply the 
                                                     h0               h                                          h0                   h
                                       derivative of cos(x) at  x   . 
                                      L’Hôspital’s Rule  
                           
                           
                          Multiple Choice  
                           
                          1.  (calculator not allowed)  
                                                  4n2
                                    lim 2                        is 
                                    nnn10000
                           
                           (A) 0 
                           (B) 1   
                                           2500
                           (C) 1 
                           (D) 4 
                           (E) nonexistent 
                           
                          2.  (calculator not allowed)  
                                               tan3(xhx) tan3
                           The                                                    is 
                                        lim
                                         h0                  h
                              
                           (A) 0 
                           (B)  2                             
                                          3sec (3x)
                           (C)  2                         
                                          sec (3x)
                           (D)                            
                                          3cot(3x)
                           (E) nonexistent 
                           
                          		
                          Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org                                 	           1 
                          3.	 (calculator not allowed)   
                                           7sxx in
                                  lim      2                    
                                 x0 xxsin(3 )
                           (A) 6 
                           (B) 2 
                           (C) 1 
                           (D) 0 
                          		
                          	
                           
                          4.  (calculator not allowed)  
                                                                                                          2
                                      x 3                                              fx()            xx,3
                           At  , the function given by                                                                       is 
                                                                                                   69xx, 3
                           
                           (A) undefined. 
                                (B)  continuous but not differentiable. 
                                (C)  differentiable but not continuous. 
                                (D)  neither continuous nor differentiable. 
                                (E)  both continuous and differentiable 
                           
                            
                          5.	(calculator allowed)  
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
                                The figure above shows the graph of a function                                         with domain                           .  Which of the 
                                                                                                                   f                          04x
                                following statements are true?  
                                              lim fx( )
                             I.                                exists  
                                              x2
                                                lim fx( )
                             II.                                 exists  
                                               x2
                                                 lim fx( )
                             III.                                exists  
                                                 x2
                           
                                (A)  I only     (B) II only      (C)  I and II only                                  (D) I and III only                  (E) I, II, and III 
                           
                           
                          		
                          Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org                                2