Explanation Given: • lim x → c ( a x + b ) = a c + b . • a , b a n d c are arbitrary constants. lim x → c f ( x ) = L is true for any function f ( x ) which is defined on an open interval around x = c , and L is a real number. That means for every number ε > 0 there exists some number δ > 0 such that if 0 < | x − c | < δ there exists | f ( x ) − L | < ε . Calculation: Description: Step 1 : Write the given function and limit. f ( x ) = a x + b . lim x → c f ( x ) = lim x → c ( a x + b ) = a c + b . Step 1 : Write | f ( x ) − L | < ε for the given function and L value. Here in the given problem f ( x ) = a x + b , L = a c + b | f ( x ) − L | = | f ( x ) − ( a c + b ) | Step 2 : Substitute f ( x ) = a x + b in the above equation and do the operation on the left side of the equation alone. | f ( x ) − ( a c + b ) | = | ( a x + b ) − ( a c + b ) | Step 3 : Remove the parenthesis for further simplification. | f ( x ) − ( a c + b ) | = | a x + b − a c − b | = | a x − a c | Step 4 : After doing multiplication, take the common term out of the absolute value. At this step we got the right side of the equation from the left side of the equation. Thereby the given equation is shown as true. | f ( x ) − ( a c + b ) | = | a ( x − c ) | = a | x − c | Step 5 : We have to show that if 0 < | x − c | < δ , then | f ( x ) − ( a c + b ) | < ε

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 2.9, Problem 16E

To determine

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Rigorously the given statement for the given values of a, b and c

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Chapter 2.9, Problem 15E

Chapter 2.9, Problem 17E

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Chapter 2 Solutions

Calculus - Standalone book

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 5PQ Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E

Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.2 - Prob. 1PQ Ch. 2.2 - Prob. 2PQ Ch. 2.2 - Prob. 3PQ Ch. 2.2 - Prob. 4PQ Ch. 2.2 - Prob. 5PQ Ch. 2.2 - Prob. 6PQ Ch. 2.2 - Prob. 7PQ Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - Prob. 13E Ch. 2.2 - 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