1-3 Continuity, End Behavior, and Limits
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1 Determine whether each function is continuous at the given x-value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. 1. f (x) = ; at x = 5 Find f ( 5). The function is defined at x = 5. Find. Construct a table that shows values of f (x) for x-values approaching 5 from the left and from the x f(x) Because, f (x) is continuous as x = 5. esolutions Manual - Powered by Cognero Page 1
2 2. f (x) = ; at x = 8 Find f (8). The function is defined at x = 8. Find. Construct a table that shows values of f (x) for x-values approaching 8 from the left and from the x f(x) Because, f (x) is continuous as x = ; at x = 5 and x = 5 Find h( 5). The function is undefined at x = 5. Find x. Construct a table that shows values of h(x) for x-values approaching 5 from the left and from the h(x) esolutions Manual - Powered by Cognero Page 2
3 Because at x = 5., but h( 5) is undefined, h(x) is discontinuous at x = 5 and has a removable discontinuity Find h(5). The function is defined at x = 5. Find. Construct a table that shows values of h(x) for x-values approaching 5 from the left and from the x h(x) Because, h(x) is continuous as x = g(x) = ; at x = 2 and x = 2 Find g(2). esolutions Manual - Powered by Cognero Page 3
4 The function is defined at x = 2. Find. Construct a table that shows values of g(x) for x-values approaching 2 from the left and from the x g(x) Because, g(x) is continuous as x = 2. Find g( 2). The function is undefined at x = 2. Find. Construct a table that shows values of g(x) for x-values approaching 2 from the left and from the x g(x) esolutions Manual - Powered by Cognero Page 4
5 Because g( 2) is undefined and g(x) approaches as x approaches 2 from the left and as x approaches 2 from the right, g(x) is discontinuous at x = 2 and has an infinite discontinuity at x = ; at x = 0 and x = 6 Find h(0). The function is undefined at x = 0. Find x. Construct a table that shows values of h(x) for x-values approaching 0 from the left and from the Because h(0) is undefined and h(x) approaches as x approaches 0 from both sides, h(x) is discontinuous at x = 0 and has an infinite discontinuity at x = 0. Find h(6). h(x) , , The function is defined at x = 6. Find. Construct a table that shows values of h(x) for x-values approaching 6 from the left and from the x h(x) esolutions Manual - Powered by Cognero Page 5
6 . Because, h(x) is continuous as x = 6. Determine between which consecutive integers the real zeros of each function are located on the given interval. 15. f (x) = 2x 4 3x 3 + x 2 3; [ 3, 3] Make a table of values for [ 3, 3]. x f (x) Because f ( 1) is positive and f (0) is negative, f (x) must have a zero between 1 and 0. Because f (1) is negative and f(2) is positive, f (x) must have a zero between 1 and 2. Notice that while there is no sign change between f (0) and f (1), f (x) does decrease as x approaches 0 from the left and then begins to increase at x = 1. Therefore, we need to check the interval more closely. The graph confirms that there are no zeros between 0 and 1. esolutions Manual - Powered by Cognero Page 6
7 17. f (x) = 3x 3 6x 2 2x + 2; [ 2, 4] Make a table of values for [ 2, 4]. x f (x) Because f ( 1) is negative and f (0) is positive, f (x) must have a zero between 1 and 0. Because f (0) is positive and f(1) is negative, f (x) must have a zero between 0 and 1. Because f (2) is negative and f (3) is positive, f (x) must have a zero between 2 and 3. The graph supports this conclusion. esolutions Manual - Powered by Cognero Page 7
8 19. ; [ 2, 4] Make a table of values for [ 2, 4]. x h(x) Notice that while there is no sign change between h(0) and h(1), h(x) does increase as x approaches 0 from the left and then begins to decrease at x = 1. Therefore, we need to check the interval more closely. The graph confirms that there are no zeros between 0 and 1. It also confirms that there are no zeros on [ 2, 4]. esolutions Manual - Powered by Cognero Page 8
9 61. CHALLENGE Determine the values of a and b so that f is continuous. To ensure continuity of the piecewise function, make sure that the function equals the same value where the parameters for x meet. These are located at x = 3 and x = 3 for this function. When x = 3, x 2 + a must equal bx + a. When x = 3, bx + a must equal. esolutions Manual - Powered by Cognero Page 9
10 GRAPHING CALCULATOR Graph each function. Analyze the graph to determine whether each function is even, odd, or neither. Confirm algebraically. If odd or even, describe the symmetry of the graph of the function. 67. h(x) = Compare h(x), h( x), and h(x) to determine if h(x) is even, odd, or neither. h( x) = h(x), so the function is even and the graph of the function is symmetric with respect to the y-axis. esolutions Manual - Powered by Cognero Page 10
11 68. f (x) = Compare f (x), f ( x), and f (x) to determine if f (x) is even, odd, or neither. The function is neither even nor odd. esolutions Manual - Powered by Cognero Page 11
12 69. g(x) = x 5 5x 3 + x Compare g(x), g( x), and g(x) to determine if g(x) is even, odd, or neither. 70. The function is odd, and the graph of the function is symmetric with respect to the origin. State the domain of each function. The domain is restricted to when the denominator does not equal 0. Therefore, x cannot equal 1 or 2. esolutions Manual - Powered by Cognero Page 12
13 71. The domain is restricted to when the denominator does not equal Therefore, x cannot equal 1 or 1 +. The domain is restricted to when g(a) 0. a must be between and esolutions Manual - Powered by Cognero Page 13
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