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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.7b
Chapter 2, Problem 2.7b

Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

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1
Recognize that the problem involves finding the limit of a trigonometric expression as \( x \) approaches \( \frac{\pi}{4} \).
Substitute \( x = \frac{\pi}{4} \) into the expression \( \cos 2x / (\cos x - \sin x) \) to check if it results in an indeterminate form.
Use the double angle identity for cosine: \( \cos 2x = 2\cos^2 x - 1 \) to rewrite the numerator.
Simplify the expression \( \cos x - \sin x \) in the denominator using trigonometric identities, if possible.
Evaluate the limit by substituting \( x = \frac{\pi}{4} \) again after simplification, ensuring the expression is no longer indeterminate.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this context, we are interested in evaluating the limit of the expression as x approaches π/4. Understanding limits is crucial for determining the value of functions at points where they may not be explicitly defined.
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Trigonometric Functions

Trigonometric functions, such as cosine and sine, are essential in calculus for analyzing periodic phenomena. In this problem, we are dealing with cos(2x), cos(x), and sin(x), which require knowledge of their properties and values at specific angles, particularly π/4. Familiarity with these functions helps in simplifying and evaluating the limit.
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Introduction to Trigonometric Functions

L'Hôpital's Rule

L'Hôpital's Rule is a technique used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. When direct substitution in the limit leads to such forms, this rule allows us to differentiate the numerator and denominator separately. Applying L'Hôpital's Rule may be necessary in this problem if the limit yields an indeterminate form upon evaluation.
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Related Practice
Textbook Question

Determine the following limits.


b. limx→2−1x(x−2){\(\displaystyle\]\lim\)_{x\(\to\)2^{-}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

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Textbook Question

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^+ h(x)

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Textbook Question

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

Textbook Question

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=x2−3x+6f\(\left\)(x\(\right\))=\(\frac{x^2-3}{x+6}\)

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Textbook Question

Complete the following sentences in terms of a limit.


b. A function is continuous from the right at a if _____ .

Textbook Question

Use the graph of gg in the figure to find the following values or state that they do not exist. <IMAGE>

limx→0g(x){\(\displaystyle\]\lim\)_{x\(\to\)0}g\(\left\)(x\(\right\))}