12950159-Exam-1-Solutions

12950159-Exam-1-Solutions - Math 135 Business Calculus Exam...

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Business Calculus Spring 2009 Exam 1 Solutions 1. (20 points) The graph of a function f is shown. (a) Determine the value of each of the following limits, if it exists. If it does not exist, explain why not. (i) lim x →− 2 f ( x ) As x approaches 2 from either side, the func- tion values get closer and closer to 2, so lim x →− 2 f ( x ) = 2 . (ii) lim x →− 1 f ( x ) As x approaches 1 from the left, the function values get unboundedly large, so the left-hand limit is lim x →− 1 f ( x ) = . As x approaches 1 from the right, the function values get unboundedly large negative, so the right- hand limit is lim x →− 1 + f ( x ) = −∞ . It follows that the two-sided limit lim x →− 1 f ( x ) does not exist. –5 –4 –3 –2 –1 1 2 3 4 5 x –5 –4 –3 –2 –1 1 2 3 4 5 y (iii) lim x 2 f ( x ) and lim x 2 + f ( x ) As x approaches 2 from the left, the function values approach 2, so the left-hand limit is lim x 2 f ( x ) = 2 . As x approaches 2 from the right, the function values approach 3, so the right-hand limit is lim x 2 + f ( x ) = 3 . (iv) lim x 2 f ( x ) Since the one-sided limits lim x 2 f ( x ) and lim x 2 + f ( x ) are not equal, then the two-sided limit lim x 2 f ( x ) does not exist. (b) Determine whether or not the function is continuous at each of the following numbers. If it is continuous, explain why it is using the deFnition. If is not continuous, explain why it is not using the deFnition. (i) a = 2 ±or a = 2, the function value is f ( 2) = 1 while the limit as x approaches 2 is lim x →− 2 f ( x ) = 1. Since lim x →− 2 f ( x ) 6 = f ( 2) , then f is not continuous at 2. (ii)
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12950159-Exam-1-Solutions - Math 135 Business Calculus Exam...

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