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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

Theory and Examples


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

255
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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]

309
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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⁺

274
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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.

226
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Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


a. ƒ(x) = tan x

266
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Textbook Question

Finding Limits Graphically


Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2


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a. Find limx→2+ f(x), limx→2− f(x), and f(2).

314
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