Answered: sec e 18. lim

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.4: Logarithmic Functions

Problem 44E

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Question

Trigonometric limits Use Theorem 3.10 to evaluate the following limits.

Expert Solution

Step 1

We have to find the limit:

$\lim_{θ \to 0} \frac{s e c θ - 1}{θ}$

We know the identity,

$\begin{array}{rcl} \cos θ s e c θ & = & 1 \\ \Rightarrow & s e c θ = \frac{1}{\cos θ} \end{array}$

Substituting above value in the limit,

$\begin{array}{rcl} \lim_{θ \to 0} \frac{s e c θ - 1}{θ} & = & \lim_{θ \to 0} \frac{\frac{1}{\cos θ} - 1}{θ} \\ = & \lim_{θ \to 0} \frac{\frac{1 - \cos θ}{\cos θ}}{θ} \\ = & \lim_{θ \to 0} \frac{1 - \cos θ}{θ \cos θ} \end{array}$

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