Prove the following limit using the limit c im,(x³ – x²) =+∞
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Class: Mathematical Analysis/
![**Proof of Limit Using the Limit Definition**
**Problem Statement:**
Prove the following limit using the limit definition:
\[
\lim_{x \to \infty} (x^3 - x^2) = +\infty
\]
**Analysis:**
To prove this limit using the definition, we must show that for every positive number \( M \), there exists a positive number \( N \) such that for all \( x > N \), \( x^3 - x^2 > M \).
**Proof Outline:**
1. **Simplify the Expression:**
- Consider the function \( f(x) = x^3 - x^2 \).
- Factor the expression: \( f(x) = x^2(x - 1) \).
2. **Determine Bounds:**
- As \( x \to \infty \), both \( x^2 \) and \( x \) grow without bound.
- Therefore, \( x^3 - x^2 = x^2(x - 1) \) grows without bound as \( x \to \infty \).
3. **Find Suitable N:**
- For \( x > 1 \), \( x - 1 > 0 \).
- Choose \( N \) such that \( x^2(x - 1) > M \).
- Simplification suggests \( x^3 - x^2 = x^2 (x - 1) > M \).
4. **Conclusion:**
- For any given \( M \), select \( N \) such that \( N > \max(1, \sqrt{M}) \).
- Thus, for all \( x > N \), \( x^2(x - 1) > M \).
This concludes that:
\[
\lim_{x \to \infty} (x^3 - x^2) = +\infty
\]
This proof effectively demonstrates the behavior of the function as \( x \) approaches infinity and confirms the limit using the formal definition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbaa9c753-6b68-41b4-8837-61aa8aa0a19b%2Fb141beb5-8610-4e1b-9859-46a2ae17ecce%2Fd5q0e9l_processed.png&w=3840&q=75)
Transcribed Image Text:**Proof of Limit Using the Limit Definition**
**Problem Statement:**
Prove the following limit using the limit definition:
\[
\lim_{x \to \infty} (x^3 - x^2) = +\infty
\]
**Analysis:**
To prove this limit using the definition, we must show that for every positive number \( M \), there exists a positive number \( N \) such that for all \( x > N \), \( x^3 - x^2 > M \).
**Proof Outline:**
1. **Simplify the Expression:**
- Consider the function \( f(x) = x^3 - x^2 \).
- Factor the expression: \( f(x) = x^2(x - 1) \).
2. **Determine Bounds:**
- As \( x \to \infty \), both \( x^2 \) and \( x \) grow without bound.
- Therefore, \( x^3 - x^2 = x^2(x - 1) \) grows without bound as \( x \to \infty \).
3. **Find Suitable N:**
- For \( x > 1 \), \( x - 1 > 0 \).
- Choose \( N \) such that \( x^2(x - 1) > M \).
- Simplification suggests \( x^3 - x^2 = x^2 (x - 1) > M \).
4. **Conclusion:**
- For any given \( M \), select \( N \) such that \( N > \max(1, \sqrt{M}) \).
- Thus, for all \( x > N \), \( x^2(x - 1) > M \).
This concludes that:
\[
\lim_{x \to \infty} (x^3 - x^2) = +\infty
\]
This proof effectively demonstrates the behavior of the function as \( x \) approaches infinity and confirms the limit using the formal definition.
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