Explanation Given: lim x → 0 x sin 1 x = 0 Key concepts applied: Formal definition of a limit: 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 ) = x sin 1 x . lim x → c f ( x ) = lim x → 0 x sin 1 x = 0 . Step 1 : Write | f ( x ) − L | < ε for the given function and L value. Here in the given problem f ( x ) = x sin 1 x , L = 0 | f ( x ) − L | = | f ( x ) − 0 | Step 2 : Substitute f ( x ) = x sin 1 x in the above equation and do the operation on the left side of the equation alone. | f ( x ) − 0 | = | x sin 1 x − 0 | = | x sin 1 x | Step 3 : Remove the parenthesis for further simplification. | f ( x ) − 0 | = | x sin 1 x | | f ( x ) − 0 | = | x | ⋅ | sin 1 x | Step 4 : Define the limits of δ

4th Edition

ISBN: 9781319323394

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 26E

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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 25E

Chapter 2.9, Problem 27E

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Chapter 2 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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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