Find the limit of the trigonometric function. (sin x)(1 – cos x)lim

To find the limit of trignometric function: limx→0sinx1-cosxx2. We know that, 1-cos2x=2sin2x And…

Answered: (sin x)(1 – cos x) lim

Math

Calculus

(sin x)(1 – cos x) lim

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.1: Verifying Trigonometric Identities

Problem 47E

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

Find the limit of the trigonometric function.

Expert Solution

Step 1

To find the limit of trignometric function: $\lim_{x \to 0} \frac{(\sin x) (1 - \cos x)}{x^{2}}$ .

We know that,

$1 - \cos 2 x = 2 \sin^{2} x$

And $\lim_{x \to 0} (\frac{\sin x}{x}) = 1$ .

Using the above formula, evaluating the limit.

$\begin{array}{rcl} \lim_{x \to 0} \frac{(\sin x) (1 - \cos x)}{x^{2}} & = & \lim_{x \to 0} \frac{(\sin x) \cdot 2 \sin^{2} (\frac{x}{2})}{x^{2}} \\ = & \lim_{x \to 0} (\frac{\sin x}{x}) \cdot \frac{2 \sin^{2} (\frac{x}{2})}{{(\frac{x}{2})}^{2} \cdot 4} \\ = & \frac{1}{2} \lim_{x \to 0} (\frac{\sin x}{x}) \cdot {(\lim_{x \to 0} \frac{\sin (\frac{x}{2})}{\frac{x}{2}})}^{2} \\ = & \frac{1}{2} \cdot 1 \cdot 1 \\ = & \frac{1}{2} \end{array}$

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