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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 14: Q. 54 (page 883)

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

Short Answer

Expert verified

The limit is $2$

Step by step solution

01

Given information

Given function $\lim_{x 鈫� 0} \frac{\sin (2 x)}{x}$

02

Evaluating the limit by using double angle formula

Using double angle formula, we get

$\lim_{x 鈫� 0} \frac{\sin (2 x)}{x} = \lim_{x 鈫� 0} \frac{2 \sin x \cos x}{x} \lim_{x 鈫� 0} \frac{\sin (2 x)}{x} = 2 \lim_{x 鈫� 0} \frac{\sin x}{x} . \lim_{x 鈫� 0} \cos x \lim_{x 鈫� 0} \frac{\sin (2 x)}{x} = 2.1.1 \lim_{x 鈫� 0} \frac{\sin (2 x)}{x} = 2$

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Most popular questions from this chapter

In Problems 43-52, find the limit as $x$ approaches $c$ of the average rate of change of each function from $c$ to $x$ .

$f (x) = \frac{1}{x^{2}}$

If we only approach c from one side, then we have a(n) ___________limit.

What are the domain and range of $f (x) = \ln x$ ?

In Problem, graph each function. Use the graph to find the indicated limit, if it exists.

$\underset{x 鈫� 0}{l i m} f (x), f (x) = \{\begin{matrix} x \\ 1 \\ 3 x \end{matrix} \begin{matrix} i f, x < 0 \\ i f, x = 0 \\ i f, x > 0 \end{matrix}$

In Problems 7鈥� 42, find each limit algebraically.

$\lim_{x 鈫� 1} \sqrt{5 x + 4}$ .

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