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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.3.5
Chapter 2, Problem 2.3.5

Centering Intervals About a Point


In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.


a=4/9, b=4/7, c=1/2

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1
First, understand the problem: We need to find a value of δ such that the interval (a, b) is centered around the point c, and for any x in this interval, the condition 0 < |x - c| < δ holds true.
Identify the given values: a = 4/9, b = 4/7, and c = 1/2. These values represent the endpoints of the interval and the center point, respectively.
Calculate the distance from c to each endpoint of the interval. This involves finding |c - a| and |c - b|. These distances will help determine the maximum allowable δ.
Compute |c - a|: Convert the fractions to a common denominator if necessary, and find the absolute difference between c and a.
Compute |c - b|: Similarly, convert the fractions to a common denominator if necessary, and find the absolute difference between c and b. The value of δ will be the smaller of these two distances, ensuring that the interval (a, b) is centered around c with the condition 0 < |x - c| < δ.

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Key Concepts

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

Intervals

An interval is a set of real numbers that contains all numbers between any two numbers in the set. In this context, the interval (a, b) represents all values x such that a < x < b. Understanding how to represent and visualize intervals on the number line is crucial for determining the relationship between the point c and the interval endpoints.
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Delta (δ) in Limits

In calculus, δ (delta) is used to define the precision of a limit. Specifically, it represents a distance from a point c such that if x is within δ of c (i.e., |x - c| < δ), then x is also within the interval (a, b). Finding an appropriate δ ensures that the values of x remain within the specified interval, which is essential for limit definitions and continuity.
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Absolute Value

The absolute value of a number measures its distance from zero on the number line, disregarding direction. In the expression |x - c|, it indicates how far x is from the point c. This concept is fundamental in determining the conditions under which x remains within the interval (a, b) when considering values close to c.
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Related Practice
Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limt→0 2t / tan t

Textbook Question

Limits and Continuity

Repeat the instructions of Exercise 1 for


1 , x ≤ ―1

1/x , 0 < |x| < 1

ƒ(x) = { 0, x = 1 ,

1 , x > 1 .

Textbook Question

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim x → π/6 √(csc² x + 5√3 tan x)

Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→3 (3x − 7) = 2

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (1 − cos 3x) / 2x

Textbook Question

Limits and Continuity


Graph the function


1 , x ≤ ―1

―x , ―1 < x < 0

ƒ(x) = { 1 , x = 0 ,

―x , 0 < x < 1

1 , x ≥ 1


Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = ―1 , 0 , and 1. Are any of the discontinuities removable? Explain.