sin no73. Investigate limnumerically for several positive integer valuesof n. Then guess the value in general.

Given: The limit is limθ→0sinnθθ. For n = 1. limθ→0sinθθ=sin00=00 Apply L'Hospital rule.…

Answered: sin no 73. Investigate lim numerically…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 82E

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Question

sin no
73. Investigate lim
numerically for several positive integer values
of n. Then guess the value in general.

Expert Solution

Step 1

Given:

The limit is $\lim_{θ \to 0} \frac{\sin n θ}{θ}$ .

For n = 1.

$\begin{array}{rcl} \lim_{θ \to 0} \frac{\sin θ}{θ} & = & \frac{\sin 0}{0} \\ = & \frac{0}{0} \end{array}$

Apply L'Hospital rule.

$\begin{array}{rcl} \lim_{θ \to 0} \frac{\sin θ}{θ} & = & \lim_{θ \to 0} \frac{\frac{d}{d θ} \sin θ}{\frac{d}{d θ} θ} \\ = & \lim_{θ \to 0} \frac{\cos θ}{1} \\ = & \frac{\cos 0}{1} = \frac{1}{1} \\ = & 1 \end{array}$

Step 2

For n = 2.

$\begin{array}{rcl} \lim_{θ \to 0} \frac{\sin 2 θ}{θ} & = & \frac{\sin 0}{0} \\ = & \frac{0}{0} \end{array}$

Apply L'Hospital rule.

$\begin{array}{rcl} \lim_{θ \to 0} \frac{\sin 2 θ}{θ} & = & \lim_{θ \to 0} \frac{\frac{d}{d θ} \sin 2 θ}{\frac{d}{d θ} θ} \\ = & \lim_{θ \to 0} \frac{2 \cos 2 θ}{2} \\ = & \frac{2 \cos 0}{2} \\ = & \frac{2 (1)}{2} \\ = & 1 \end{array}$

For n = 3.

$\begin{array}{rcl} \lim_{θ \to 0} \frac{\sin 3 θ}{θ} & = & \frac{\sin 0}{0} \\ = & \frac{0}{0} \end{array}$

Apply L'Hospital rule.

$\begin{array}{rcl} \lim_{θ \to 0} \frac{\sin 3 θ}{θ} & = & \lim_{θ \to 0} \frac{\frac{d}{d θ} \sin 3 θ}{\frac{d}{d θ} θ} \\ = & \lim_{θ \to 0} \frac{3 \cos 3 θ}{3} \\ = & \frac{3 \cos 0}{3} \\ = & \frac{3 (1)}{3} \\ = & 1 \end{array}$