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31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) – sin a = h h/2 | sin x Then use the inequality < 1 for x # 0 to show that |sin(a + h) – sin a| < |h| for all a. Finally, prove rigorously that lim sin x = sin a.

31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) – sin a = h h/2 | sin x Then use the inequality < 1 for x # 0 to show that |sin(a + h) – sin a| < |h| for all a. Finally, prove rigorously that lim sin x = sin a.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) – sin a = h h/2 | sin x Then use the inequality < 1 for x # 0 to show that |sin(a + h) – sin a| < |h| for all a. Finally, prove rigorously that lim sin x = sin a.
Transcribed Image Text:31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) – sin a = h h/2 | sin x Then use the inequality < 1 for x # 0 to show that |sin(a + h) – sin a| < |h| for all a. Finally, prove rigorously that lim sin x = sin a.
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