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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![**Understanding Limits and Derivatives: A Practical Example**
This example discusses a limit that represents the derivative of a function \(f(x)\) at a specific point \(c\). We are given the following limit to analyze and determine \(f(x)\) and \(c\).
\[ \lim_{{x \to 4}} \frac{4 \sqrt{x} - 8}{x - 4} \]
### Step-by-Step Solution
To solve for \(f(x)\) and \(c\), recognize the limit definition of a derivative, \(f'(x)\), which is generally given by:
\[ f'(c) = \lim_{{x \to c}} \frac{f(x) - f(c)}{x - c} \]
In this scenario, comparing with the given limit expression, we can see:
1. **Identify \(c\)**:
- \(c = 4\), as the limit approaches \(x = 4\).
2. **Identifying \(f(x)\)**:
- The numerator \(4 \sqrt{x} - 8\) matches the form \(f(x) - f(c)\).
- Therefore, \(f(x) = 4 \sqrt{x}\).
- To find \(f(4)\):
\[ f(4) = 4 \sqrt{4} = 4 \cdot 2 = 8 \]
So, the limit confirms that the function \(f(x)\) and the point \(c\) are as follows:
#### Final Results:
- Function \(f(x)\):
\[ f(x) = 4 \sqrt{x} \]
- Point \(c\):
\[ c = 4 \]
In this analysis, no graphs or diagrams were provided. The solution primarily revolves around understanding how to identify parts of the limit expression that correspond to the function \(f(x)\) and the point \(c\) it is evaluated at.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbba488a1-bd87-42aa-8e5a-6cda657e400a%2Fe4e105ca-d4c8-42c1-b389-41658ae8a601%2Ftmvveyp_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Limits and Derivatives: A Practical Example**
This example discusses a limit that represents the derivative of a function \(f(x)\) at a specific point \(c\). We are given the following limit to analyze and determine \(f(x)\) and \(c\).
\[ \lim_{{x \to 4}} \frac{4 \sqrt{x} - 8}{x - 4} \]
### Step-by-Step Solution
To solve for \(f(x)\) and \(c\), recognize the limit definition of a derivative, \(f'(x)\), which is generally given by:
\[ f'(c) = \lim_{{x \to c}} \frac{f(x) - f(c)}{x - c} \]
In this scenario, comparing with the given limit expression, we can see:
1. **Identify \(c\)**:
- \(c = 4\), as the limit approaches \(x = 4\).
2. **Identifying \(f(x)\)**:
- The numerator \(4 \sqrt{x} - 8\) matches the form \(f(x) - f(c)\).
- Therefore, \(f(x) = 4 \sqrt{x}\).
- To find \(f(4)\):
\[ f(4) = 4 \sqrt{4} = 4 \cdot 2 = 8 \]
So, the limit confirms that the function \(f(x)\) and the point \(c\) are as follows:
#### Final Results:
- Function \(f(x)\):
\[ f(x) = 4 \sqrt{x} \]
- Point \(c\):
\[ c = 4 \]
In this analysis, no graphs or diagrams were provided. The solution primarily revolves around understanding how to identify parts of the limit expression that correspond to the function \(f(x)\) and the point \(c\) it is evaluated at.
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