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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. 53 (page 883)

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

Short Answer

Expert verified

The limit equals $1$

Step by step solution

01

Given information

Given function $\lim_{x 鈫� 0} \frac{\tan x}{x}$

02

Evaluating the limit by breaking into sine and cosine components

Calculating, we get

$\lim_{x 鈫� 0} \frac{\tan x}{x} = \lim_{x 鈫� 0} (\frac{\sin x}{x} 路 \frac{1}{\cos x}) \lim_{x 鈫� 0} \frac{\tan x}{x} = \lim_{x 鈫� 0} (\frac{\sin x}{x}) 路 \lim_{x 鈫� 0} (\frac{1}{\cos x}) \lim_{x 鈫� 0} \frac{\tan x}{x} = 1 路 \frac{1}{1} \lim_{x 鈫� 0} \frac{\tan x}{x} = 1$

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