Explanation Procedure used: “L’ Hospital’s rule: 1: Check whether the lim x → a f ( x ) g ( x ) is an indeterminate form 0 0 or ± ∞ ± ∞ . 2: If so obtain the derivative of f ( x ) and g ( x ) , separately. 3: Compute the lim x → a f ′ ( x ) g ′ ( x ) , if this limit exist then lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) . 4 If it does not exist repeat the step 2 and step 3.” Calculation: The given limit is lim x → 0 ( e 3 x x 2 − 1 x 2 − 3 x ) . Compute the common denominator as, ( e 3 x x 2 − 1 x 2 − 3 x ) = e 3 x x 2 − 1 x 2 − 3 x ( x x ) = e 3 x x 2 − 1 x 2 − 3 x x 2 = e 3 x − 1 − 3 x x 2 Thus, rewrite the given limit as lim x → 0 ( e 3 x − 1 − 3 x x 2 ) . Step 1: Obtain the value of lim x → 0 ( e 3 x − 1 − 3 x x 2 ) . lim x → 0 ( e 3 x − 1 − 3 x x 2 ) = ( e 3 ( 0 ) − 1 − 3 ( 0 ) x 2 ) = 1 − 1 0 = 0 0 It is observed to be an indeterminate form. Thus, apply L’Hospital’s rule. Step 2: It is noticed that f ( x ) = ( e 3 x − 1 − 3 x ) and g ( x ) = x 2 Obtain the derivative of f ( x )

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 12, Problem 63RE

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The value of the given limit after computing a common denominator.

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Chapter 12, Problem 62RE

Chapter 12, Problem 64RE

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Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for "correct") if the argument is valid, or enter | (for "incorrect") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.) ☐ 1. For all n > 1, seriesΣ In(n) In(n) converges. 2, 1, arctan(n) the series arctan(n) n³ ☐ 4. For all n > 1, 123 converges. 1 n ln(n) series In(n) diverges. 2n . and the seriesΣconverges, so by the Comparison Test, 2, 3, and the series converges, so by the Comparison Test, the series-3 1 converges. ☐ 6. For all n > 2, In(n) >, and the series Σ converges, so by the Comparison Test, the seriesΣ In(n) converges.

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Chapter 12 Solutions

Calculus with Applications (11th Edition)

Ch. 12.1 - Find the first four terms of the sequence having...Ch. 12.1 - Prob. 2YT Ch. 12.1 - Prob. 3YT Ch. 12.1 - Prob. 4YT Ch. 12.1 - Prob. 5YT Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - List the first n terms of the geometric sequence...Ch. 12.1 - Prob. 5E

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