교과서 질문
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→−3 (x² − 9) / (x + 3) = −6
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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
2장, 문제 2.5.27
At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
05:34
Intro to Continuity
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.
추천 영상:
5:57Graphs of Common Functions
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.
추천 영상:
5:10Finding the Domain and Range of a Graph
관련 실천
교과서 질문
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→−5⁻ (3x) / (2x + 10)
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교과서 질문
Theory and Examples
Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.
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교과서 질문
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) = −∞
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교과서 질문
Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find
b. limx→c 2f(x)g(x)
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교과서 질문
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.
376
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