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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 4e
Chapter 2, Problem 4e

Limits and Continuity


Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.


e. x + ƒ(x)

Verified step by step guidance
1
Understand that the limit of a sum of functions as x approaches a certain value is the sum of the limits of those functions. This is a fundamental property of limits.
Identify the individual limits given in the problem: lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2.
Recognize that the function in question is x + ƒ(x). We need to find lim (x → 0) [x + ƒ(x)].
Apply the property of limits: lim (x → 0) [x + ƒ(x)] = lim (x → 0) x + lim (x → 0) ƒ(x).
Since lim (x → 0) x = 0 (as x approaches 0, the value of x itself approaches 0), combine this with the given limit of ƒ(x) to find the overall limit: lim (x → 0) [x + ƒ(x)] = 0 + 1/2.

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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. In this context, understanding how to evaluate limits as x approaches 0 for the functions ƒ(x) and g(x) is crucial. The limit helps determine the behavior of functions near specific points, which is essential for solving the given problem.
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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. This concept is important when analyzing the behavior of ƒ(x) and g(x) as x approaches 0. Continuity ensures that small changes in x result in small changes in the function's output, allowing for straightforward limit evaluation.
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Intro to Continuity

Algebra of Limits

The algebra of limits refers to the rules that govern how limits can be combined, such as the sum, difference, product, and quotient of limits. In this problem, knowing that the limit of a sum is the sum of the limits allows us to find the limit of the function x + ƒ(x) as x approaches 0 by simply adding the limits of x and ƒ(x). This principle simplifies the process of finding limits for more complex functions.
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Related Practice
Textbook Question

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


c. limx→1 f(x) does not exist.


344
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Textbook Question

Limits and Continuity


Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.


f. [ƒ(x) • cos x ] / x―1

255
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Textbook Question

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


d. limx→c f(x) exists at every point c in (-1,1).


301
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Textbook Question

Limits and Continuity

In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.


lim (x lim g(x)) = 2

x→-4 x→0

188
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Textbook Question

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


b. limx→2 f(x)=2


369
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Textbook Question

Limits and Continuity


In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.


lim ((4―g(x)) / x ) = 1

x→0

275
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