Textbook Question
Which of the following statements about the function y=f(x) graphed here are true, and which are false?
i. f(0)=1
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
Chapter 2, Problem 3h
Limits and Continuity
Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.
h. 1 / ƒ(t)
Verified step by step guidance1
Identify the given limits: lim t → t₀ ƒ(t) = -7 and lim t → t₀ g(t) = 0. We need to find lim t → t₀ of 1/ƒ(t).
Recall the limit property for the reciprocal function: If lim t → t₀ ƒ(t) = L and L ≠ 0, then lim t → t₀ 1/ƒ(t) = 1/L.
Apply the property to the given function: Since lim t → t₀ ƒ(t) = -7 and -7 ≠ 0, we can use the reciprocal limit property.
Substitute the limit value into the reciprocal property: lim t → t₀ 1/ƒ(t) = 1/(-7).
Conclude that the limit of 1/ƒ(t) as t approaches t₀ is the reciprocal of -7, which is -1/7.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit describes the value that a function approaches as the input approaches a certain point. In this context, the limit of ƒ(t) as t approaches t₀ is given as -7, indicating that as t gets closer to t₀, ƒ(t) gets closer to -7. Understanding limits is crucial for analyzing the behavior of functions near specific points, especially when dealing with continuity and discontinuity.
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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. In this case, since lim t → t₀ ƒ(t) = -7, we can infer that ƒ(t) is continuous at t₀ if ƒ(t₀) is also -7. Continuity is essential for ensuring that limits can be evaluated without encountering undefined behavior.
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Reciprocal Limits
The limit of the reciprocal of a function, such as 1/ƒ(t), can be evaluated using the limit of the function itself. If lim t → t₀ ƒ(t) = -7, then lim t → t₀ (1/ƒ(t)) = 1/(-7) = -1/7, provided that ƒ(t) does not approach zero. This concept is important for understanding how limits behave under operations like taking reciprocals.
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Related Practice
Textbook Question
Limits and Continuity
Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.
f. | ƒ(t) |
Textbook Question
Which of the following statements about the function y=f(x) graphed here are true, and which are false?
a. limx→2 f(x) does not exist.
Textbook Question
Limits and Continuity
Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.
e. cos (g(t))
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
Textbook Question
Which of the following statements about the function y=f(x) graphed here are true, and which are false?
g. limx→1 f(x) does not exist.
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