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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 2.7a
Chapter 2, Problem 2.7a

Limits and Continuity
On what intervals are the following functions continuous?


a. ƒ(x) = x¹/³

Verified step by step guidance
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.
Identify the type of function: The function ƒ(x) = x^(1/3) is a root function, specifically a cube root function.
Consider the domain of the function: Cube root functions are defined for all real numbers, meaning there are no restrictions on x for ƒ(x) = x^(1/3).
Analyze the behavior of the function: Since cube root functions do not have any discontinuities like jumps, holes, or vertical asymptotes, they are continuous everywhere in their domain.
Conclude the intervals of continuity: Based on the analysis, ƒ(x) = x^(1/3) is continuous on the interval (-∞, ∞), which includes all real numbers.

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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. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is crucial for determining where a function does not have breaks, jumps, or asymptotes.
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Intro to Continuity

Limits

Limits describe the behavior of a function as it approaches a certain point from either side. Understanding limits is essential for analyzing continuity, as a function can only be continuous if the limit exists and matches the function's value at that point. This concept helps in identifying points of discontinuity in a function.
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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 redefining it. Jump discontinuities happen when the left-hand and right-hand limits exist but are not equal, while infinite discontinuities occur when a function approaches infinity at a point. Recognizing these types is vital for analyzing the continuity of functions.
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Related Practice
Textbook Question

[Technology Exercise] Roots


Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.


a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .

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Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


h(t)=cot t


a. [π/4,3π/4]

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Textbook Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.


lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as


a. x → 0⁺

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Textbook Question

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let g(x) = (x² − 2) / (x − √2)


a. Make a table of the values of g at the points x=1.4,1.41,1.414, and so on through successive decimal approximations of √2. Estimate limx→√2 g(x).

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Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


g(x)=x²−2x


a. [1, 3]

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Textbook Question

Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find


a. limx→−2 (p(x) + r(x) + s(x))

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