Explanation Given: o Adapt the argument in Example 2 o Limit: lim x → c x 2 = c 2 for all c 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 . Compare the given limit lim x → c x 2 = c 2 with general definition of limit lim x → c f ( x ) = L and write 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 . By the definition, if we prove that for any value of 0 < | x − c | < δ there exists | f ( x ) − L | < ε then the limit is vigorously true. Calculation: Description: Step 1 : Write the given function and consider this as f ( x ) . lim x → c x 2 = c 2 . Step 2 : Compare the given limit lim x → c x 2 = c 2 with general definition of limit lim x → c f ( x ) = L and write the corresponding values of f ( x ) , c , a n d L . f ( x ) = x 2 c = c L = c 2 Step 3 : By the formal definition of limit, define the values of ε a n d δ . ε > 0 δ > 0 Step 4 : By the definition, if we prove that for any value of 0 < | x − c | < δ there exists | f ( x ) − L | < ε then the given limit exists and equals the given value which is vigorously true. For the given problem, 0 < | x − c | < δ becomes as 0 < | x − c | < δ . For the given problem, | f ( x ) − L | < ε becomes as | x 2 − c 2 | < ε

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 2.9, Problem 19E

To determine

To Prove:

Rigorously that the given limit exists

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Chapter 2.9, Problem 18E

Chapter 2.9, Problem 20E

Chapter 2 Solutions

CALCULUS (CLOTH)

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

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. 2.2 - 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 - Prob. 14E Ch. 2.2 - Prob. 15E Ch. 2.2 - Prob. 16E Ch. 2.2 - Prob. 17E Ch. 2.2 - Prob. 18E Ch. 2.2 - Prob. 19E Ch. 2.2 - 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To Prove: Rigorously that the given limit exists

Solution Summary: The author explains that the given limit exists for all c and compares it with the formal definition of limit.