   Chapter 3.3, Problem 57E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle θ. Find lim θ → 0 +   s d To determine

To find: The value of limθ0+sd.

Explanation

Given:

The circular arc of length is s and the chord of length is d.

Formula used:

Circular arc length: s = rθ, where r is the radius of the circle.

Result used:

The value of limit limx0sinxx=1.

Calculation:

The given picture is shown in the below Figure 1.

From Figure 1, TQ=QR=d2 and RPQ=θ2.

Calculate the value of d.

From the triangle PQR as shown in Figure 1,

sinθ2=opposite sidehypotenusesinθ2=d2rrsinθ2=d22rsinθ2=d

Therefore, the value is d=2rsinθ2.

Consider limθ0+sd

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