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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 2.7b
Chapter 2, Problem 2.7b

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


b. g(x) = x³/⁴

Verified step by step guidance
1
Identify the type of function: The function g(x) = x^(3/4) is a power function, which is generally continuous wherever it is defined.
Determine the domain of the function: Since the exponent 3/4 is a positive rational number, g(x) is defined for all x ≥ 0. This is because the fourth root of a negative number is not real.
Check for any points of discontinuity: For power functions like g(x) = x^(3/4), there are no discontinuities within their domain. Therefore, the function is continuous on its entire domain.
Conclude the intervals of continuity: Since g(x) is defined and continuous for all x ≥ 0, the interval of continuity is [0, ∞).
Summarize the findings: The function g(x) = x^(3/4) is continuous on the interval [0, ∞), as it is a power function with a domain of all non-negative real numbers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions, especially at points where they may not be explicitly defined. They help determine continuity and differentiability, which are essential for evaluating functions over intervals.
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One-Sided Limits

Continuity

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 vital for understanding how functions behave and ensuring that there are no breaks, jumps, or holes in their graphs.
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Intro to Continuity

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Analyzing these functions requires checking continuity at the boundaries of the pieces to ensure that the function behaves consistently. Understanding how to evaluate these functions across their defined intervals is essential for determining where they are continuous.
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Piecewise Functions
Related Practice
Textbook Question

Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.


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b. limt→4−(t−⌊t⌋)

Textbook Question

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


lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as


a. x → 0⁺

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(t)=2+cos t


b. [0,π]

Textbook Question

Estimating Limits


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


Let f(x) = (x² - 9) / (x + 3)


b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.

Textbook Question

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?



b. limx→2 f(x) does not exist

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]