Explanation Given: lim x → 1 x 3 = 1 Key concepts applied: Formal definition of a limit: Let f ( x ) be a function defined on an open interval containing c (except possibly at c ) and let L be a real number. The statement lim x → c f ( x ) = L means that for every ε > 0 , there exists δ > 0 , such that if 0 < | x − c | < δ , then | f ( x ) − L | < ε . The following steps may be followed to rigorously prove a given limit using the formal definition of the given limit. First, write the given function and consider this as f ( x ) , then compare the given limit with the general definition of limit lim x → c f ( x ) = L and note down the corresponding values of f ( x ) , c a n d L . By the formal definition of limit, define the values of ε a n d δ > 0 . To show that for any value of 0 < | x − c | < δ there exist | f ( x ) − L | < ε , we need a way of relating | x − c | < δ and | f ( x ) − L | < ε . Start with the ε term and algebraically manipulate it until a relation between ε a n d δ is obtained, or something of the form | g ( x ) | ⋅ | x − c | < ε is obtained, where g ( x ) is some function of x , in which case restrict δ (which forces a restriction on x ) so that you can get an upper bound for g ( x ) . Use this upper bound for g ( x ) to find δ . So if we algebraically get to and g ( x ) < B for some number B , then if we can make B | x − c | < ε then this will force | g ( x ) | ⋅ | x − c | to be less than ε because | g ( x ) | ⋅ | x − c | < B | x − c | < ε . We can then relate ε a n d δ > 0 which can then be used to relate | x − c | < δ and | f ( x ) − L | < ε , thus completing the proof. Calculation: Description lim x → 1 x 3 = 1 Step 1: Write the given function and consider this as f ( x ) f ( x ) = x 3 Step 2: Compare the given limit lim x → 1 x 3 = 1 with general definition of limit lim x → c f ( x ) = L and write the corresponding values of f ( x ) , c a n d L

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 2.9, Problem 23E

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Use the formal definition of the limit to prove the given statement rigorously.

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Chapter 2.9, Problem 22E

Chapter 2.9, Problem 24E

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Chapter 2 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

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

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Use the formal definition of the limit to prove the given statement rigorously.

Solution Summary: The author explains how to rigorously prove a given limit using the formal definition of the limit.

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Chapter 2 Solutions