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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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Finding Area Between Curves that Cross on the Interval

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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One-Sided Limits

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

Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = x², x = -2

310
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Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limt→0 sin(1 − cos t) / (1 − cos t)

310
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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 .

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Textbook Question

Theory and Examples


a. If limx→0 f(x) / x² = 1, find limx→0 f(x).

255
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Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x→∞ (2√x + x⁻¹) / (3x − 7)

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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.

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