Using a diagram of the unit circle and the Pythagorean Theorem, show that Conclude that sin2

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Using a diagram of the unit circle and the Pythagorean Theorem, show that

sin 0 < (1- cos 0) + sin 0  0

Conclude that sin2 θ ≤ 2(1 − cos θ) ≤ θ2 and use this to give an alternative proof that the limit in Exercise 51 equals 0. Then give an alternative proof of the result in Exercise 54.


Data From Exercise 51

Explain why  lim(csc cot 0) 00involves an indeterminate form, and then prove that the limit equals 0.


Data From Exercise 54

Investigate lim h0 1 - cos h numerically or graphically. Then prove that the limit is equal to 1/2. See the proof of Theorem 2.

THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1| and lim 0-0 1- cos 0 0 = 0

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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