Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Provide an appropriate response.
![### Problem Statement
Given the inequality for a function \( f(x) \):
\[ x^3 \leq f(x) \leq x \quad \text{for } x \text{ in } [-1, 1], \]
find the limit of \( f(x) \) as \( x \) approaches 0, if it exists.
\[ \lim_{x \to 0} f(x) \]
### Approach
To solve for the limit, we can use the Squeeze Theorem. According to the Squeeze Theorem, if \( g(x) \leq f(x) \leq h(x) \) for all \( x \) in an interval \( I \) containing \( c \), except possibly at \( c \) itself, and
\[ \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L, \]
then
\[ \lim_{x \to c} f(x) = L. \]
In this problem, let us identify \( g(x) \) and \( h(x) \):
\[ g(x) = x^3 \quad \text{and} \quad h(x) = x. \]
Now, evaluate the limits of \( x^3 \) and \( x \) as \( x \) approaches 0:
\[ \lim_{x \to 0} x^3 = 0 \]
\[ \lim_{x \to 0} x = 0 \]
Since \( x^3 \leq f(x) \leq x \) and both functions approach 0 as \( x \) approaches 0, by the Squeeze Theorem:
\[ \lim_{x \to 0} f(x) = 0. \]
### Conclusion
The limit of \( f(x) \) as \( x \) approaches 0 is \( 0 \).
\[ \boxed{0} \]
This is confirmed by the application of the Squeeze Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75d9c7d3-d734-453a-85e0-780b79a7d6f6%2F0774de59-9928-4cba-8b40-46c48431f1f7%2Fh4nmyeb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Given the inequality for a function \( f(x) \):
\[ x^3 \leq f(x) \leq x \quad \text{for } x \text{ in } [-1, 1], \]
find the limit of \( f(x) \) as \( x \) approaches 0, if it exists.
\[ \lim_{x \to 0} f(x) \]
### Approach
To solve for the limit, we can use the Squeeze Theorem. According to the Squeeze Theorem, if \( g(x) \leq f(x) \leq h(x) \) for all \( x \) in an interval \( I \) containing \( c \), except possibly at \( c \) itself, and
\[ \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L, \]
then
\[ \lim_{x \to c} f(x) = L. \]
In this problem, let us identify \( g(x) \) and \( h(x) \):
\[ g(x) = x^3 \quad \text{and} \quad h(x) = x. \]
Now, evaluate the limits of \( x^3 \) and \( x \) as \( x \) approaches 0:
\[ \lim_{x \to 0} x^3 = 0 \]
\[ \lim_{x \to 0} x = 0 \]
Since \( x^3 \leq f(x) \leq x \) and both functions approach 0 as \( x \) approaches 0, by the Squeeze Theorem:
\[ \lim_{x \to 0} f(x) = 0. \]
### Conclusion
The limit of \( f(x) \) as \( x \) approaches 0 is \( 0 \).
\[ \boxed{0} \]
This is confirmed by the application of the Squeeze Theorem.
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