To determine Find the limit by using L ‘Hospital’s rule. Explanation Given information: The function is lim t → 0 t ( 1 − cos t ) t − sin t . L ‘Hospital’s rule: Suppose that f ( a ) = g ( a ) = 0 that f and g are differentiable on an open interval I containing a, and that g ′ ( x ) ≠ 0 on I if x ≠ a then, lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) Assuming that the limit on the right side of this equation exists. Calculation: Show the function as follows: lim t → 0 t ( 1 − cos t ) t − sin t Substitute 0 for t . lim t → 0 t ( 1 − cos t ) t − sin t = 0 ( 1 − cos 0 ) 0 − sin 0 = 0 0 Consider f ( t ) = t ( 1 − cos t ) f ( t ) = t ( 1 − cos t ) = t − t cos t Differentiate f ( t ) with respect to t . f ′ ( t ) = d d t ( t − t cos t ) = ( d d t t ) − d d t ( t cos t ) = 1 − [ t d d t ( cos t ) + cos t ( d d t t ) ] = 1 − ( t ( − sin t ) + cos t × 1 ) = 1 − cos t + t sin t Consider g ( t ) = t − sin t Differentiate g ( t ) with respect to t . g ′ ( t ) = d d t ( t − sin t ) = 1 − cos t Apply the L Hospital rule as follows: lim t → 0 f ( t ) g ( t ) = lim t → 0 f ′ ( t ) g ′ ( t ) (1) Substitute ( 1 − cos t + t sin t ) for f ′ ( t ) and ( 1 − cos t ) for g ′ ( t ) in Equation (1). lim t → 0 f ( t ) g ( t ) = lim t → 0 1 − cos t + t sin t 1 − cos t = 1 − cos 0 + 0 sin 0 1 − cos 0 = 0 0 Differentiate f ′ ( t ) with respect to t

University Calculus: Early Transcendentals, Books a la Carte Edition (3rd Edition)

3rd Edition

ISBN: 9780321999610

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 4.5, Problem 23E

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Chapter 4.5, Problem 22E

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Chapter 4 Solutions

University Calculus: Early Transcendentals, Books a la Carte Edition (3rd Edition)

Ch. 4.1 - In Exercises 16, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 3E Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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