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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![**Problem Statement**
Find the limit:
\[
\lim_{{x \to -3^-}} \frac{1}{{x + 3}}
\]
**Solution Explanation**
The problem asks us to find the left-hand limit of the function \(f(x) = \frac{1}{x + 3}\) as \(x\) approaches \(-3\).
**Conceptual Explanation**
- **Left-Hand Limit**: This refers to approaching the point from the left side on a number line. In this context, as \(x\) approaches \(-3\) from the left, the expression \(x + 3\) becomes a very small negative number.
- **Behavior of the Function**: As \(x + 3\) approaches zero from the negative side, \(\frac{1}{x + 3}\) diverges towards negative infinity. This is because dividing 1 by a very small negative number yields a very large negative value.
- **Conclusion**: Therefore, the left-hand limit of the function as \(x\) approaches \(-3\) is negative infinity.
Overall, understanding these types of limits involves recognizing the behavior of rational functions as their denominators approach zero from different directions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55f59351-6cab-453e-87b8-764174a74643%2Fd03c7786-a262-49d6-8fd7-aa613712b2a9%2Fdvn300r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the limit:
\[
\lim_{{x \to -3^-}} \frac{1}{{x + 3}}
\]
**Solution Explanation**
The problem asks us to find the left-hand limit of the function \(f(x) = \frac{1}{x + 3}\) as \(x\) approaches \(-3\).
**Conceptual Explanation**
- **Left-Hand Limit**: This refers to approaching the point from the left side on a number line. In this context, as \(x\) approaches \(-3\) from the left, the expression \(x + 3\) becomes a very small negative number.
- **Behavior of the Function**: As \(x + 3\) approaches zero from the negative side, \(\frac{1}{x + 3}\) diverges towards negative infinity. This is because dividing 1 by a very small negative number yields a very large negative value.
- **Conclusion**: Therefore, the left-hand limit of the function as \(x\) approaches \(-3\) is negative infinity.
Overall, understanding these types of limits involves recognizing the behavior of rational functions as their denominators approach zero from different directions.
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