Q. 56 In Problems 53-56, use the prope... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 14: Q. 56 (page 883)

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

Short Answer

Expert verified

The limit is $1$

Step by step solution

Given information

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

Express in standard form and calculate

Calculating, we get

$\lim_{x 鈫� 0} \frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}} = \frac{\sin x}{x} (\frac{\sin x}{x}) + \frac{\sin x}{x} (\frac{\cos x - 1}{x}) \lim_{x 鈫� 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = \lim_{x 鈫� 0} (\frac{\sin x}{x} (\frac{\sin x}{x}) + \frac{\sin x}{x} (\frac{\cos x - 1}{x})) \lim_{x 鈫� 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = \lim_{x 鈫� 0} (\frac{\sin x}{x}) 路 \lim_{x 鈫� 0} (\frac{\sin x}{x}) + \lim_{x 鈫� 0} (\frac{\sin x}{x}) 路 \lim_{x 鈫� 0} (\frac{\cos x - 1}{x}) \lim_{x 鈫� 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = 1 (1) + 1 (0) \lim_{x 鈫� 0} (\frac{\sin^{2} x + \sin x (\cos x - 1)}{x^{2}}) = 1$

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91影视

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

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

Geometry

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Statistics

Study anywhere. Anytime. Across all devices.

91影视

In Problems 53-56, use the properties of limits and the facts thatlimx鈫�0sinxx=1limx鈫�0cosx-1x=0limx鈫�0sinx=0limx鈫�0cosx=1limx鈫�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

Geometry

Discrete Mathematics

Logic and Functions

Decision Maths

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

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