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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 83b
Chapter 2, Problem 83b

A sine limit It can be shown that 1−x^2/ 6 ≤ sin x/ x ≤1, for x near 0.
Use these inequalities to evaluate lim x→0 sin x/ x.

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Start by understanding the given inequality: \(1 - \frac{x^2}{6} \leq \frac{\sin x}{x} \leq 1\) for \(x\) near 0. This inequality is derived from the Taylor series expansion of \(\sin x\).
Recognize that the limit we want to evaluate is \(\lim_{x \to 0} \frac{\sin x}{x}\).
Apply the Squeeze Theorem, which states that if \(f(x) \leq g(x) \leq h(x)\) for all \(x\) in some interval around \(c\) (except possibly at \(c\)), and \(\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L\), then \(\lim_{x \to c} g(x) = L\).
In this case, set \(f(x) = 1 - \frac{x^2}{6}\), \(g(x) = \frac{\sin x}{x}\), and \(h(x) = 1\). Notice that both \(f(x)\) and \(h(x)\) approach 1 as \(x\) approaches 0.
Conclude that by the Squeeze Theorem, \(\lim_{x \to 0} \frac{\sin x}{x} = 1\). This is because both the lower bound \(1 - \frac{x^2}{6}\) and the upper bound 1 converge to 1 as \(x\) approaches 0.

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Key Concepts

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

Squeeze Theorem

The Squeeze Theorem is a fundamental concept in calculus that states if a function is 'squeezed' between two other functions that converge to the same limit at a certain point, then the squeezed function must also converge to that limit. In this case, the inequalities provided serve as the two bounding functions for sin(x)/x as x approaches 0.
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Finding Global Extrema (Extreme Value Theorem)

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. Evaluating limits helps in understanding the continuity and behavior of functions at specific points, particularly where they may not be explicitly defined, such as sin(x)/x at x=0.
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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. Understanding the properties and behavior of these functions, especially near critical points like 0, is essential for evaluating limits involving trigonometric expressions, such as sin(x)/x.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→∞ cot^−1

Textbook Question

Use an appropriate limit definition to prove the following limits.


lim x→1 (5x−2) =3;

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Textbook Question

Suppose f(x) = {x^2 − 5x + 6 / x − 3 if x≠3

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Textbook Question

a. Use the Intermediate Value Theorem to show that the equation has a solution in the given interval.


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Textbook Question

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

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Textbook Question

Suppose g(x) = {x^2−5x if x≤−1

ax^3−7 if x>−1.

Determine a value of the constant a for which lim x→−1 g(x) exists and state the value of the limit, if possible.