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 = b − n . 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 − 0 | < ε whenever n > N . That is, | b − n − 0 | < ε whenever n > N . Solve and obtain the value of n

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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A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 10.2, Problem 104E

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To prove: The limit of the sequence {b−n} is 0 for b>1 using the formal definition of the limit of the sequence.

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Chapter 10.2, Problem 103E

Chapter 10.2, Problem 105E

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(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

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Let f be defined as follows. y = f(x) = x² - 5x (a) Find the average rate of change of y with respect to x in the following intervals. from x = 4 to x = 5 from x = 4 to x = 4.5 from x = 4 to x = 4.1 (b) Find the (instantaneous) rate of change of y at x = 4. Need Help? Read It Master It

Chapter 10 Solutions

Calculus: Early Transcendentals (3rd Edition)

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To prove: The limit of the sequence { b − n } is 0 for b > 1 using the formal definition of the limit of the sequence.

Solution Summary: The author explains that the limit of the sequence leftb-nright is 0 for b>1.

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To prove: The limit of the sequence {b−n} is 0 for b>1 using the formal definition of the limit of the sequence.

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To prove: The limit of the sequence { b − n } is 0 for b &gt; 1 using the formal definition of the limit of the sequence.

Solution Summary: The author explains that the limit of the sequence leftb-nright is 0 for b>1.

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To prove: The limit of the sequence {b−n} is 0 for b>1 using the formal definition of the limit of the sequence.

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Chapter 10 Solutions

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To prove: The limit of the sequence { b − n } is 0 for b > 1 using the formal definition of the limit of the sequence.