Explanation Given information: The equivalent statement that shows that the sine function possess Direct Substitution propertyis , lim x → a sin x = sin a . A modified version is if h = x − a , then x = a + h and as x tends to a this implies h tends to 0 that is lim h → 0 sin ( a + h ) = sin a . Formula used: Addition of trigonometric functions, sin ( c + d ) = sin c cos d + cos c sin d Sum Rule for Limits, lim x → p ( m ( x ) + n ( x ) ) = lim x → p m ( x ) + lim x → p n ( x ) Limit of trigonometric functions, lim θ → 0 sin ( θ ) = 0 and lim θ → 0 cos ( θ ) = 1 Proof: It is provided that the equivalent statement that shows that the sine function possess Direct Substitution propertyis lim x → a sin x = sin a . A modified version is if h = x − a , then x = a + h and as x tends to a this implies h tends to 0 that is lim h → 0 sin ( a + h ) = sin a . Consider the term, sin ( a + h ) . Apply the Addition of trigonometric functions, sin ( c + d ) = sin c cos d + cos c sin d That is, sin ( a + h ) = sin a cosh + cos a sinh Apply the limit function lim h → 0 on both the sides of the above equation

15th Edition

ISBN: 9781133109631

Author: Stewart, JAMES, Day, Troy

Publisher: Cengage Learning,

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Chapter 2.4, Problem 47E

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To prove: The sine function possess Direct Substitution property.

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Chapter 2.4, Problem 46E

Chapter 2.4, Problem 48E

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Chapter 2 Solutions

Biocalculus

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