Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
a. limx→0+ (1 − cos x) / |cos x − 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 2.5.6a
Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
<IMAGE>
a. Does f (1) exist?
Verified step by step guidance1
Identify the piecewise function definition for f(x) at x = 1. According to the problem, f(x) is defined as 1 when x = 1.
To determine if f(1) exists, check if there is a specific value assigned to f(x) at x = 1 in the piecewise function.
Since the piecewise function explicitly defines f(x) = 1 for x = 1, f(1) exists and is equal to 1.
In general, for a function value to exist at a specific point, the function must be defined at that point in the piecewise definition.
Conclude that f(1) exists because the piecewise function provides a specific value for f(x) at x = 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Piecewise Functions
A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has distinct formulas for various intervals of x, which means that the output depends on which interval the input falls into. Understanding how to evaluate piecewise functions is crucial for determining function values at specific points.
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05:36
Piecewise Functions
Function Existence
For a function to exist at a certain point, it must have a defined output for that input. This involves checking if the input falls within the domain of the function and if the corresponding expression yields a real number. In the context of the question, we need to evaluate f(1) to see if it is defined according to the piecewise conditions.
Recommended video:
03:07Cases Where Limits Do Not Exist
Continuity at a Point
Continuity at a point means that the function's value at that point matches the limit of the function as it approaches that point from both sides. While the question specifically asks about the existence of f(1), understanding continuity helps in analyzing the behavior of the function around that point, which can be important for further questions regarding limits or derivatives.
Recommended video:
05:34Intro to Continuity
Related Practice
Textbook Question
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .
Textbook Question
Theory and Examples
a. If limx→0 f(x) / x² = 1, find limx→0 f(x).
Textbook Question
Limits and Continuity
On what intervals are the following functions continuous?
a. ƒ(x) = x¹/³
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let g(x) = (x² − 2) / (x − √2)
a. Make a table of the values of g at the points x=1.4,1.41,1.414, and so on through successive decimal approximations of √2. Estimate limx→√2 g(x).
Textbook Question
Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find
a. limx→−2 (p(x) + r(x) + s(x))