Explanation Given: The graph of the function f ( x ) = 2 − 1 x is given below and if 0 < | x − 1 | < δ , then | f ( x ) − 1 | < ε Here ε = 0.05 , 0.01 , and 0.005 . Refer to figure in question. Calculation: i) If ε = 0.05 and 0 < | x − 1 | < δ , then | f ( x ) − 1 | < 0.05 Now replace f ( x ) with 2 − 1 x : | f ( x ) − 1 | < 0.05 ⇒ | 2 − 1 x − 1 | < 0.05 ⇒ | 1 − 1 x | < 0.05 ⇒ | 1 x − 1 | < 0.05 ⇒ − 0.05 < 1 x − 1 < 0.05 ⇒ 1 − 0.05 < 1 x < 1 + 0.05 ⇒ 0.95 < 1 x < 1.05 ⇒ 95 100 < 1 x < 105 100 ⇒ 100 95 > x > 100 105 ⇒ 100 95 − 1 > x − 1 > 100 105 − 1 ⇒ 5 95 > x − 1 > − 5 105 − − − − ( 1 ) Let δ be 5 105 . Then 0 < | x − 1 | < δ becomes 0 < | x − 1 | < 5 105 : ⇒ − 5 105 < x − 1 < 5 105 , which satisfies inequality (1). Hence, δ = 5 105 is the required value. ii) If ε = 0.01 and 0 < | x − 1 | < δ , then | f ( x ) − 1 | < 0.01 . Now replace f ( x ) with 2 − 1 x : | f ( x ) − 1 | < 0.01 ⇒ | 2 − 1 x − 1 | < 0.01 ⇒ | 1 − 1 x | < 0.01 ⇒ | 1 x − 1 | < 0.01 ⇒ − 0.01 < 1 x − 1 < 0.01 ⇒ 1 − 0.01 < 1 x < 1 + 0.01 ⇒ 0

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ISBN: 9780357092477

Author: Larson

Publisher: Cengage

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Chapter 1.2, Problem 38E

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To calculate: The value of δ using the graph of the function f(x)=2−1x which is given in the question and if 0<|x−1|<δ, then |f(x)−1|<ε.

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Chapter 1.2, Problem 37E

Chapter 1.2, Problem 39E

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The value of δ using the graph of the function f ( x ) = 2 − 1 x which is given in the question and if 0 < | x − 1 | < δ , then | f ( x ) − 1 | < ε .

Solution Summary: The author explains the function f(x)=2-1x, which is given in the question.

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To calculate: The value of δ using the graph of the function f(x)=2−1x which is given in the question and if 0<|x−1|<δ, then |f(x)−1|<ε.

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Chapter 1 Solutions

The value of δ using the graph of the function f ( x ) = 2 − 1 x which is given in the question and if 0 &lt; | x − 1 | &lt; δ , then | f ( x ) − 1 | &lt; ε .

Solution Summary: The author explains the function f(x)=2-1x, which is given in the question.

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To calculate: The value of δ using the graph of the function f(x)=2−1x which is given in the question and if 0<|x−1|<δ, then |f(x)−1|<ε.

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Chapter 1 Solutions

The value of δ using the graph of the function f ( x ) = 2 − 1 x which is given in the question and if 0 < | x − 1 | < δ , then | f ( x ) − 1 | < ε .