Textbook Question
At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 2, Problem 2.5.2
In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?
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Verified step by step guidance1
Step 1: Understand the definition of continuity. A function is continuous on an interval if it is continuous at every point in that interval. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.
Step 2: Identify the interval of interest, which is [-1, 3]. This means we need to check the continuity of the function at every point within this interval, including the endpoints -1 and 3.
Step 3: Examine the graph of the function over the interval [-1, 3]. Look for any breaks, jumps, or holes in the graph, as these indicate points of discontinuity.
Step 4: Check the endpoints of the interval. Ensure that the function is defined at x = -1 and x = 3, and that the limits from the left and right at these points match the function's value at these points.
Step 5: Identify any specific points within the interval where the function might not be continuous. This could be due to a jump discontinuity, an infinite discontinuity, or a removable discontinuity. Note these points and the reasons for discontinuity, if any.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Continuity of Functions
A function is continuous on an interval if it is defined at every point in that interval and does not have any breaks, jumps, or holes. Specifically, for a function to be continuous at a point, the limit of the function as it approaches that point must equal the function's value at that point.
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Types of Discontinuities
Discontinuities can be classified into three main types: removable, jump, and infinite. A removable discontinuity occurs when a function is not defined at a point but can be made continuous by defining it appropriately. A jump discontinuity happens when the left-hand and right-hand limits at a point do not match, while an infinite discontinuity occurs when the function approaches infinity at a certain point.
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Evaluating Limits
To determine continuity, evaluating limits is essential. The limit of a function as it approaches a point from both sides must exist and be equal to the function's value at that point. If the limits differ or do not exist, the function is discontinuous at that point, which can help identify where the function fails to be continuous.
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Related Practice
Textbook Question
Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 3x − 1 = 0
Textbook Question
In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)
lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞
Textbook Question
Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find
b. limx→c 2f(x)g(x)
Textbook Question
Using the Sandwich Theorem
a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?
Give reasons for your answer.
[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.
Textbook Question
Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.