Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.5.27

Chapter 2, Problem 2.5.27

At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³

Verified step by step guidance

Step 1: Understand the concept of continuity. 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 type of function given. The function y = (2x - 1)^(1/3) is a cube root function, which is continuous everywhere in its domain.

Step 3: Determine the domain of the function. The cube root function is defined for all real numbers, so the domain of y = (2x - 1)^(1/3) is all real numbers.

Step 4: Since the function is defined for all real numbers and cube root functions are continuous over their entire domain, y = (2x - 1)^(1/3) is continuous for all x in the real number set.

Step 5: Conclude that the function y = (2x - 1)^(1/3) is continuous everywhere on the real number line, meaning it does not have any points of discontinuity.

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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 at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.

Cube Root Function

The cube root function, denoted as y = (x)^(1/3), is defined for all real numbers. Unlike square roots, which are only defined for non-negative numbers, cube roots can take any real number as input, including negative values. This characteristic ensures that the cube root function is continuous everywhere on the real number line.

Domain of the Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function y = (2x – 1)^(1/3), the expression inside the cube root can take any real number, meaning the domain is all real numbers. Understanding the domain is crucial for determining where the function is continuous.

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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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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.)

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Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find

b. limx→c 2f(x)g(x)

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.

At what points are the functions in Exercises 13–30 continuous?y = (2x – 1)¹/³

Verified video answer for a similar problem:

Key Concepts

Continuity of Functions

Cube Root Function

Domain of the Function

At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³