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Lesson 4 One-Sided Limits Example Find xlim 4 x4 Based on the discussion and work in Lesson 3, most students say that lim x 4 0 . x4 However, xlim 4 Notation: x 4 = DNE (Does Not Exist). x a means x approaches a from the right-hand side of a . Hence, x > a. x a means x approaches a from the left-hand side of a . Hence, x < a. Left-hand side of a x a Right-hand side of a x a a Definition (Right-Hand Limit) Let f be a function defined on the interval (a, a + r), where r > 0. (Note: the function f is defined on the right-hand side of x = a.) If as x a , f ( x ) M , then we write lim f ( x ) M . xa Definition (Left-Hand Limit) Let f be a function defined on the interval (a r, a), where r > 0. (Note: the function f is defined on the left-hand side of x = a.) If as x a , f ( x ) N , then we write lim f ( x ) N . xa Examples Find the following limits, if they exist. 1. lim ( 4 x 5 ) x 2 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 We need to be able to approach 2 from the right-hand side. Thus, the function y 4 x 5 needs to be defined on some open interval of the form ( 2 , 2 r ) , where r > 0, so that we can approach 2 from the right-hand side. Since the function y 4 x 5 is a polynomial, then its domain is the set of all real numbers. Thus, we can approach 2 from the right-hand side. Since as x 2 , 4 x 5 3 , then lim ( 4 x 5 ) 8 5 3 . x2 Answer: 3 Animation of this limit. NOTE: The limit as x approaches 2 from the left-hand side would also be 3. ( 4x 5 ) 3 . That is, xlim 2 2. lim ( 2w 2 1) w3 We need to be able to approach 3 from the left-hand side. Thus, the 2 function y 2w 1 needs to be defined on some open interval of the form ( 3 r , 3 ) , where r > 0, so that we can approach 3 from the left-hand 2 side. Since the function y 2w 1 is a polynomial, then its domain is the set of all real numbers. Thus, we can approach 3 from the right-hand side. 2 2 Since as w 3 , 2w 1 19 , then lim ( 2w 1) 19 . w3 Answer: 19 Animation of this limit. NOTE: The limit as w approaches 3 from the right-hand side would also 2 be 19. That is, lim ( 2w 1) 19 . w3 3. lim t 4 2 5t t3 We need to be able to approach 4 from the right-hand side. Thus, the 2 5t y function needs to be defined on some open interval of the form t3 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 ( 4 , 4 r ) , where r > 0, so that we can approach 4 from the right-hand 2 5t side. Since the domain of the function y is the set of all real t3 numbers such that t 3 , then the function is defined on the open interval ( 4 , 3 ) . Thus, we can approach 4 from the right-hand side. Thus, lim t 4 2 5t 22 22 . t3 1 Answer: 22 NOTE: The limit as t approaches 4 from the left-hand side would also be 2 5t lim 22 . 22 . That is, t 4 t3 4. x 2 5 x 24 lim 2 x 3 4 x 7 x 15 We need to be able to approach 3 from the left-hand side. Thus, the function x 2 5 x 24 y 2 needs to be defined on some open interval of the form 4 x 7 x 15 ( 3 r , 3 ) , where r > 0, so that we can approach 3 from the left-hand side. x 2 5 x 24 Since the domain of the function y is the set of all real 4 x 2 7 x 15 5 x number such that and x 3 , then the function is defined on the 4 open interval ( 1, 3 ) . Thus, we can approach 3 from the left-hand side. Thus, ( x 3)( x 8) x 2 5 x 24 x8 11 lim lim lim = = 2 x 3 ( x 3)( 4 x 5) x 3 4 x 7 x 15 x 3 4x 5 17 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 Answer: 11 17 NOTE: The limit as x approaches 3 from the right-hand side would also be x 2 5 x 24 11 11 . That is, xlim . 3 4 x 2 7 x 15 17 17 5. lim x4 x4 We need to be able to approach 4 from the right-hand side. Thus, the function y x 4 needs to be defined on some open interval of the form ( 4 , 4 r ) , where r > 0, so that we can approach 4 from the right-hand side. Since the domain of the function y x 4 is the set of real numbers given by the interval [ 4 , ) , then the function is defined on the open interval ( 4 , ) . Thus, we can approach 4 from the right-hand side. Thus, lim x 4 x4 0 0 Answer: 0 6. lim x 4 x4 We need to be able to approach 4 from the left-hand side. Thus, the function y x 4 needs to be defined on some open interval of the form ( 4 r , 4 ) , where r > 0, so that we can approach 4 from the left-hand side. Since the domain of the function y x 4 is [ 4 , ) , then the function is not defined on any open interval of the form ( 4 r , 4 ) , where r > 0. Thus, we can NOT approach 4 from the left-hand side. Thus, lim x 4 x 4 = DNE Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 Answer: DNE Definition A neighborhood of the real number x = a is any open interval containing the real number a. Definition A deleted neighborhood of the real number x = a is any open interval containing the real number a for which the number a is removed (or deleted.) Theorem Let f be a function defined on a deleted neighborhood of x = a. Then lim f ( x ) L if and only if lim f ( x ) L and lim f ( x ) L . xa xa Example Find xlim 4 xa x4 This limit is not a one-sided limit. If a side is not specified, then you have to be able to approach the number from both sides. For the limit to exist, the two onesided limits must exist and be equal. Since lim x 4 0 by Example 5 above x4 and x lim 4 x 4 = DNE by Example 6 above, then lim x4 x 4 = DNE by the theorem above. Answer: DNE NOTE: The domain of a function plays a role in determining whether we can approach a number from the right-hand side and/or the left-hand side. Example Find tlim 5 5t The domain of the function y 0 0 and lim t 5 5t is ( , 5 ] . Thus, t lim 5 5 t = DNE. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 5t = Answer: DNE Example Find x lim 4 16 x 2 2 Sign of 16 x : + 4 The domain of the function y Thus, lim x4 4 16 x 2 is [ 4 , 4 ] . 16 x 2 = DNE. Thus, lim x4 16 x 2 = DNE Answer: DNE 16 x 2 = DNE since lim x4 NOTE: Also, we have that xlim 4 16 x 2 = DNE in the example above. Example Find tlim 7 Sign of t7 6t 11 : 4 t 7 6t 11 + + 11 6 The domain of the function y Thus, t lim 7 4 4 7 t7 11 [7, ) . is , 6t 11 6 t7 lim 6t 11 = DNE. Thus, t 7 4 t 7 6t 11 = DNE Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 Answer: DNE t7 4 lim NOTE: Also, we have that 11 6t 11 = DNE since t 6 lim 11 t 6 4 t7 6t 11 = DNE in the example above. Example Find x lim 7 Sign of 6 63x 4 9 x 6 63x 4 9 x 6 9 x 4 ( 7 x 2 ) : + 7 The domain of the function y Thus, lim x 7 6 6 63 x 4 9 x 6 is 63x 4 9 x 6 = DNE. Thus, lim x 7 0 7 , 6 + 7 7 . 63x 4 9 x 6 = DNE Answer: DNE NOTE: Also, we have that lim x 7 lim x0 x0 x 7 6 63x 4 9 x 6 = DNE since 63x 4 9 x 6 = DNE in the example above. However, 63x 4 9 x 6 0 since lim x0 6 lim 6 6 lim 6 63x 4 9 x 6 0 and 63x 4 9 x 6 0 . Another application of One-Sided Limits is to calculate the limit of a piecewise function at its breakup points. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 Examples Find the following limits, if they exist. 1. 2x 2 , x 1 f ( x) . If f ( x ) , then find xlim 1 5 x , x 1 f ( x ) and We will need to calculate the following one-sided limits xlim 1 lim f ( x ) . Since the domain of f is the set of all real numbers, we will be x 1 able to approach 1 from both the right-hand and left-hand sides. lim f ( x ) = lim ( 5 x ) 5 1 4 . Animation of this right-hand limit. x 1 x 1 x 1 x 1 f ( x ) 5 x lim f ( x ) = lim ( 2 x 2 ) 2 x 1 x 1 Animation of this left-hand limit. x 1 x 1 f ( x) 2 x 2 f ( x ) 4 and lim f ( x ) 2 . Thus, by the theorem above, Thus, xlim 1 x 1 lim f ( x ) = DNE. x 1 Answer: DNE This graph was created using Maple. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 2. x 2 4 x 15 , x 3 g ( x ) , 3 x 5 , then find lim g ( x ) . 2 x 42 If x3 2 , x5 3 x x We will need to calculate the following one-sided limits lim x 3 lim x 3 g ( x ) and g ( x ) . Since the domain of g is the set of all real numbers, we will be able to approach 3 from both the right-hand and left-hand sides. lim x 3 g ( x ) = lim x3 2 x 42 x 3 x 3 g ( x ) lim x3 6 42 36 6 2x 42 g ( x ) = lim ( x 2 4 x 15) 9 12 15 6 x3 x 3 x 3 g ( x ) x 2 4 x 15 Thus, lim x 3 g ( x ) 6 and lim g ( x ) 6 . Thus, by the theorem above, x3 lim g ( x ) 6 . x3 Answer: 6 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 g ( x ) = DNE since lim g ( x ) 10 and lim g ( x ) NOTE: xlim 5 x5 x5 3. 3 2 2 x 12 x 26 , x 2 2x 2 8 h( x ) , 2x3 x 2 3 14 2 x , x3 If , then find 52 lim h( x ) and x2 lim h( x ) . x3 We will need to calculate the following one-sided limits lim x 2 lim x 2 h( x ) and h( x ) . Since the domain of h is the set of all real numbers such that x 3 , we will be able to approach 2 from both the right-hand and lefthand sides. lim 2x 2 8 = x2 lim h( x ) = lim 2( x 2)( x 2) = x2 x 2 x 2 x 2 lim lim x 2 x 2 2( x2 4) = x2 2( x 2) 8 2x 2 8 x 2 x 2 h( x ) x2 Animation of this right-hand limit. 3 2 x 12 x 26 6 24 26 8 x2 x2 2 3 x 2 x 2 h( x ) x 2 12 x 26 2 Animation of this left-hand limit. lim h( x ) = lim Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 Thus, lim x 2 h( x ) 8 and lim x 2 h( x ) 8 . Thus, by the theorem h( x ) 8 . above, x lim 2 h( x ) and We will need to calculate the following one-sided limits xlim 3 lim h( x ) . Since the domain of h is the set of all real numbers such that x 3 x 3 , we will be able to approach 3 from both the right-hand and left-hand sides. lim h( x ) = lim x3 x 3 3 14 2 x 3 8 2 x 3 x 3 h( x ) 3 14 2 x Animation of this right-hand limit. 2x 2 8 10 2 lim h( x ) = lim x3 x3 x2 5 2x 2 8 x 3 x 3 h( x ) x2 Animation of this left-hand limit. h( x ) 2 and lim h( x ) 2 . Thus, by the theorem above, Thus, xlim 3 x3 lim h( x ) 2 . x3 Answer: 8 ; 2 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 4. lim x8 x8 x8 Since the absolute value function is a piecewise function, we will need to x8 y calculate the one-sided limits. The domain of is the set of all x8 real numbers such that x 8 . Thus, we will be able to approach 8 from both the right-hand and left-hand sides. lim x8 x8 x8 = x lim 8 x8 11 = xlim 8 x8 x 8 x 8 x 8 0 x 8 x 8 lim x8 x8 x8 = x lim 8 ( x 8) ( 1) 1 = x lim 8 x8 x 8 x 8 x 8 0 x 8 ( x 8) Thus, xlim 8 x8 x8 = DNE Answer: DNE COMMENT: In order to find the derivative of the function y x 8 at x 8 , we must calculate the limit above. We will see in a later lesson that the derivative x8 lim y x 8 x 8 of the function at is undefined since x 8 = DNE. x8 5. 25 9t 2 lim 5 3t 5 t 3 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 Since the absolute value function is a piecewise function, we will need to 25 9t 2 calculate the one-sided limits. The domain of y 3t 5 is the set of all 5 5 . Thus, we will be able to approach from 3 3 both the right-hand and left-hand sides. real numbers such that t 5 t 3 lim 5 t 3 lim 5 t 3 5 t 3 lim 5 t 3 t 5 3t 5 3t 5 0 3t 5 3t 5 3 25 9t 2 3t 5 = lim 5 t 3 25 9t 2 3t 5 = lim 5 t 3 ( 5 3t ) ( 5 3t ) = 3t 5 [ ( 3t 5 ) ] ( 5 5 ) 10 5 t 3 lim t 5 3t 5 3t 5 0 3t 5 ( 3t 5 ) 5 3t 3 25 9t 2 3t 5 = lim 5 t 3 25 9t 2 5 3t = lim 5 t 3 ( 5 3t ) ( 5 3t ) = 5 3t ( 3t 5 ) 5 5 10 25 9t 2 Thus, lim5 3t 5 = DNE t 3 Answer: DNE Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 6. w 2 11w 28 lim w7 w7 Since the absolute value function is a piecewise function, we will need to w 2 11w 28 calculate the one-sided limits. The domain of y is the w7 set of all real numbers such that w 7 . Thus, we will be able to approach 7 from both the right-hand and left-hand sides. 2 In order to find the absolute value of w 11w 28 , we need to find the 2 sign of w 11w 28 using the material from Lesson 1. Sign of w2 11w 28 ( w 4 ) (w 7 ) : + 7 2 Thus, when w 7 , w 11w 28 0 . 2 2 Thus, w 11w 28 = ( w 11w 28 ) . Thus, lim lim w 7 w7 w 2 11w 28 w7 ( w 4 ) (w 7 ) = w7 = lim w7 lim w7 ( w 2 11w 28 ) = w7 [ ( w 4)] 3 2 When w 7 , w 11w 28 0 . 2 2 Thus, w 11w 28 = w 11w 28 . Thus, lim w7 w 2 11w 28 w7 = lim w7 w 2 11w 28 = w7 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 + 4 lim w7 ( w 4 ) (w 7 ) = w7 Thus, wlim 7 lim w7 ( w 4) 3 w 2 11w 28 w7 = DNE Answer: DNE 2 COMMENT: In order to find the derivative of the function y w 11w 28 at w 7 , we must calculate the limit above. We will see in a later lesson that the 2 derivative of the function y w 11w 28 at w 7 is undefined since lim w7 w 2 11w 28 w7 = DNE. 2 What does the graph of y w 11w 28 look like? You can use the graph of 2 y w 2 11w 28 to get the graph of y w 11w 28 . First, graph y w 2 11w 28 . Leave the part of the graph that is above the horizontal waxis as it is because this is where the y-coordinates are positive. But since y w 2 11w 28 , then this is where w 2 11w 28 is positive and w 2 11w 28 = w 2 11w 28 when w2 11w 28 y is positive. Leave the w-intercept(s) as is because this is where the y-coordinate is zero and w 2 11w 28 = w 2 11w 28 when w2 11w 28 y is zero. Now, for the part of the graph that is below the horizontal w-axis, the y-coordinates are 2 2 negative. Thus, w 11w 28 = ( w 11w 28 ) when w2 11w 28 y is negative. Thus, for the part of the graph that is below the horizontal w-axis, you reflect it above the horizontal w-axis. 2 In order to graph y w 11w 28 , we will use the information above that w 2 11w 28 ( w 4 ) (w 7 ) . Thus, the w-intercepts of the graph are the points ( 7 , 0 ) and ( 4 , 0 ) . Using the fact that the vertex of the parabola is on Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 the axis of symmetry and that the axis of symmetry is w coordinate of the vertex is w 11 , then the w2 11 . Since y ( w 4 ) ( w 7 ) , then the y2 11 11 8 11 14 = coordinate of the parabola is obtained by y 2 2 2 2 2 9 9 3 3 11 , . . Thus, the vertex of the parabola is 4 4 2 2 2 2 The graph of y w 11w 28 : 2 The graph of y w 11w 28 : These graphs were created using Maple. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850 7. 2 x 2 8x lim x4 x4 Since the absolute value function is a piecewise function, we will need to 2 x 2 8x calculate the one-sided limits. The domain of y x 4 is the set of all real numbers such that x 4 . Thus, we will be able to approach 4 from both the right-hand and left-hand sides. lim x 4 2 x 2 8x = x4 lim x 4 2 x 2 8x = x4 lim x 4 2x ( x 4 ) = x4 lim 2x 8 lim ( 2x ) 8 x 4 x 4 x 4 x 4 0 x 4 x 4 lim x 4 2 x 2 8x = x4 lim x 4 2 x 2 8x = ( x 4) lim x 4 2x ( x 4 ) = ( x 4) x 4 x 4 x 4 x 4 0 x 4 ( x 4) 2 x 2 8x Thus, x lim 4 x 4 = DNE Answer: DNE Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1850