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:
\[
\lim_{{x \to -\infty}} \left( \frac{7}{x^3} - 9 \right)
\]
**Explanation**
This problem involves finding the limit of a mathematical expression as \( x \) approaches negative infinity. The expression consists of two terms: \( \frac{7}{x^3} \) and \( -9 \).
1. **Term Analysis:**
- **\(\frac{7}{x^3}\):** As \( x \to -\infty \), \( x^3 \to -\infty \) since the cube of a negative number is also negative. This causes the fraction \(\frac{7}{x^3} \to 0\) because dividing a constant number by an increasingly large negative value leads to zero.
- **\(-9\):** This is a constant term and remains unchanged as \( x \) approaches any value. It does not affect the limit of the other terms but is part of the final limit calculation.
2. **Limit Calculation:**
- Since \(\frac{7}{x^3} \to 0\), the expression simplifies to \(-9\).
- Thus, the limit of the entire expression, as \( x \to -\infty \), is \(-9\).
**Conclusion**
\[
\lim_{{x \to -\infty}} \left( \frac{7}{x^3} - 9 \right) = -9
\]
This concludes the analysis for finding the limit as \( x \) approaches negative infinity. The primary focus was on determining the behavior of the variable-dependent term and combining it with the constant to find the overall limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c6604b-039c-4afd-872a-b167ca11814a%2Fb4e271bb-4c62-4da4-bae5-bbc43fcf00c4%2Fvyijfyb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the limit:
\[
\lim_{{x \to -\infty}} \left( \frac{7}{x^3} - 9 \right)
\]
**Explanation**
This problem involves finding the limit of a mathematical expression as \( x \) approaches negative infinity. The expression consists of two terms: \( \frac{7}{x^3} \) and \( -9 \).
1. **Term Analysis:**
- **\(\frac{7}{x^3}\):** As \( x \to -\infty \), \( x^3 \to -\infty \) since the cube of a negative number is also negative. This causes the fraction \(\frac{7}{x^3} \to 0\) because dividing a constant number by an increasingly large negative value leads to zero.
- **\(-9\):** This is a constant term and remains unchanged as \( x \) approaches any value. It does not affect the limit of the other terms but is part of the final limit calculation.
2. **Limit Calculation:**
- Since \(\frac{7}{x^3} \to 0\), the expression simplifies to \(-9\).
- Thus, the limit of the entire expression, as \( x \to -\infty \), is \(-9\).
**Conclusion**
\[
\lim_{{x \to -\infty}} \left( \frac{7}{x^3} - 9 \right) = -9
\]
This concludes the analysis for finding the limit as \( x \) approaches negative infinity. The primary focus was on determining the behavior of the variable-dependent term and combining it with the constant to find the overall limit.
![Find the limit.
\[
\lim_{{x \to 6}} \frac{{-0.4}}{{x - 6}}
\]
This expression represents the limit of the function \(-0.4/(x - 6)\) as \(x\) approaches 6.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c6604b-039c-4afd-872a-b167ca11814a%2Fb4e271bb-4c62-4da4-bae5-bbc43fcf00c4%2Fccmycyih_processed.png&w=3840&q=75)
Transcribed Image Text:Find the limit.
\[
\lim_{{x \to 6}} \frac{{-0.4}}{{x - 6}}
\]
This expression represents the limit of the function \(-0.4/(x - 6)\) as \(x\) approaches 6.
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