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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. 6 (page 907)

In Problems 1鈥� 6, find each limit
$\lim_{x 鈫� \frac{蟺}{4}} \frac{\tan x}{1 + \cos^{2} x}$

Short Answer

Expert verified

The value is $\frac{2}{3}$

Step by step solution

01

Step 1. Distributing the limit

The given limit is $\lim_{x 鈫� \frac{蟺}{4}} \frac{\tan x}{1 + \cos^{2} x}$

It can be written as

$\frac{\lim_{x 鈫� \frac{蟺}{4}} \tan x}{\lim_{x 鈫� \frac{蟺}{4}} (1 + \cos^{2} x)}$

02

Step 2. Value of the limit

On simplifying the limit

$\frac{\lim_{x 鈫� \frac{蟺}{4}} \tan x}{\lim_{x 鈫� \frac{蟺}{4}} (1 + \cos^{2} x)} = \frac{\lim_{x 鈫� \frac{蟺}{4}} \tan x}{\lim_{x 鈫� \frac{蟺}{4}} 1 + \lim_{x 鈫� \frac{蟺}{4}} \cos^{2} x} = \frac{\tan (\frac{蟺}{4})}{\lim_{x 鈫� \frac{蟺}{4}} 1 + \cos^{2} (\frac{蟺}{4})} = \frac{\tan (\frac{蟺}{4})}{1 + \cos^{2} (\frac{蟺}{4})} = \frac{1}{1 + {(\frac{\sqrt{2}}{2})}^{2}} = \frac{2}{3}$

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