Explanation Definition used: “The sequence { a n } converges to L provided the terms of a n can be made arbitrarily close to L by taking n sufficiently large. That is, { a n } has the unique limit L if given any ε > 0 , it is possible to find a positive integer N (depending only on ε ) such that | a n − L | < ε whenever n > N , which can be expressed as lim n → ∞ a n = L .” Proof: Consider the sequence a n = 3 n 2 4 n 2 + 1 . Choose the positive integer N and given any ε > 0 . From the definition, there exist a real number n sufficiently large such that, | a n − 3 4 | < ε whenever n > N . That is, | 3 n 2 4 n 2 + 1 − 3 4 | < ε whenever n > N . Solve and obtain the value of n . | 12 n 2 − 3 − 12 n 2 4 ( 4 n 2 + 1 ) | < ε for n > N | − 3 4 ( 4 n 2 + 1 )

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Limits

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Chapter 10.2, Problem 103E

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To prove: The limit of the sequence {3n24n2+1} is 34 using the formal definition of the limit of the sequence.

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Chapter 10.2, Problem 102E

Chapter 10.2, Problem 104E

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Calculus: Early Transcendentals (3rd Edition)

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To prove: The limit of the sequence { 3 n 2 4 n 2 + 1 } is 3 4 using the formal definition of the limit of the sequence.

Solution Summary: The author explains that the sequence lefta_nright converges to L, provided the terms of