Chapter 14: Q. 56 (page 883)

In Problems 53-56, use the properties of limits and the facts that
$\lim_{x \to 0} \frac{\sin x}{x} = 1 \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 \lim_{x \to 0} \sin x = 0 \lim_{x \to 0} \cos x = 1$
$\lim_{x \to 0} \frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}$

Short Answer

Expert verified

The limit is $1$

Step by step solution

Given information

Given function $\lim_{x \to 0} \frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}$

Express in standard form and calculate

Calculating, we get

$\lim_{x \to 0} \frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}} = \frac{\sin x}{x} (\frac{\sin x}{x}) + \frac{\sin x}{x} (\frac{\cos x - 1}{x}) \lim_{x \to 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = \lim_{x \to 0} (\frac{\sin x}{x} (\frac{\sin x}{x}) + \frac{\sin x}{x} (\frac{\cos x - 1}{x})) \lim_{x \to 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = \lim_{x \to 0} (\frac{\sin x}{x}) \cdot \lim_{x \to 0} (\frac{\sin x}{x}) + \lim_{x \to 0} (\frac{\sin x}{x}) \cdot \lim_{x \to 0} (\frac{\cos x - 1}{x}) \lim_{x \to 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = 1 (1) + 1 (0) \lim_{x \to 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = 1$

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In Problems 53-56, use the properties of limits and the facts that
$\lim_{x \to 0} \frac{\sin x}{x} = 1 \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 \lim_{x \to 0} \sin x = 0 \lim_{x \to 0} \cos x = 1$
$\lim_{x \to 0} \frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}$

Short Answer

Step by step solution

Given information

Express in standard form and calculate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Study anywhere. Anytime. Across all devices.

In Problems 53-56, use the properties of limits and the facts thatlimx→0sinxx=1limx→0cosx-1x=0limx→0sinx=0limx→0cosx=1limx→0sin2x+sinx(cosx-1)x2

Short Answer

Step by step solution

Given information

Express in standard form and calculate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problems 53-56, use the properties of limits and the facts that
$\lim_{x \to 0} \frac{\sin x}{x} = 1 \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 \lim_{x \to 0} \sin x = 0 \lim_{x \to 0} \cos x = 1$
$\lim_{x \to 0} \frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}$