Answered: EXAMPLE 10 Trigonometric limits…

EXAMPLE 10 Trigonometric limits Evaluate the following trigonometric limits. sin? x 1 - cos 2x а. lim xSo 1 - cos x b. lim sin x

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 4E

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Question

EXAMPLE 10 Trigonometric limits Evaluate the following trigonometric limits.
sin? x
1 - cos 2x
а. lim
xSo 1 - cos x
b. lim
sin x

Expert Solution

given

$G i v e n a) \lim_{x \to 0} \frac{\sin^{2} x}{1 - \cos x} b) \lim_{x \to 0} \frac{1 - \cos 2 x}{\sin x}$

solution for a)

$a) \lim_{x \to 0} \frac{\sin^{2} x}{1 - \cos x} = \lim_{x \to 0} \frac{1 - \cos^{2} x}{1 - \cos x} (\sin c e \sin^{2} x = 1 - \cos^{2} x) = \lim_{x \to 0} \frac{1^{2} - {(\cos x)}^{2}}{1 - \cos x} = \lim_{x \to 0} \frac{(1 + \cos x) (1 - \cos x)}{1 - \cos x} (\sin c e a^{2} - b^{2} = (a + b) (a - b)) = \lim_{x \to 0} 1 + \cos x = 1 + 1 = 2 (\cos 0 = 1) T h e r e f o r e \lim_{x \to 0} \frac{\sin^{2} x}{1 - \cos x} = 2$