3. Consider the following integral: * 2 + 3x + 7 dr x4 + 2x (a) Show that the integral converges using an inequality comparison. (b) Show that the integral converges using a limit comparison.
3. Consider the following integral: * 2 + 3x + 7 dr x4 + 2x (a) Show that the integral converges using an inequality comparison. (b) Show that the integral converges using a limit comparison.
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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Thanks for any help on this! I think a and b go together, let me know if I need to resubmit it. (3)
![### Problem 3: Analyzing Convergence of an Integral
Consider the following integral:
\[
\int_{3}^{\infty} \frac{x^2 + 3x + 7}{x^4 + 2x} \, dx
\]
**Tasks:**
(a) **Inequality Comparison Method:**
- Demonstrate that the integral converges by finding a function \( g(x) \) such that \( 0 \leq \frac{x^2 + 3x + 7}{x^4 + 2x} \leq g(x) \) for all \( x \geq 3 \).
- Verify that the integral of \( g(x) \) from 3 to \(\infty\) converges.
(b) **Limit Comparison Method:**
- Use the limit comparison test to show convergence.
- Identify a function \( h(x) \) with known convergence properties.
- Compute \(\lim_{x \to \infty} \frac{\frac{x^2 + 3x + 7}{x^4 + 2x}}{h(x)}\) and use the result to conclude about the convergence of the original integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa40a52fa-ca63-40a5-a54d-3f60ff8c6b34%2Fe9fcdb54-e04d-4e90-9851-b0ee5c5db6ab%2Fp6kzcj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 3: Analyzing Convergence of an Integral
Consider the following integral:
\[
\int_{3}^{\infty} \frac{x^2 + 3x + 7}{x^4 + 2x} \, dx
\]
**Tasks:**
(a) **Inequality Comparison Method:**
- Demonstrate that the integral converges by finding a function \( g(x) \) such that \( 0 \leq \frac{x^2 + 3x + 7}{x^4 + 2x} \leq g(x) \) for all \( x \geq 3 \).
- Verify that the integral of \( g(x) \) from 3 to \(\infty\) converges.
(b) **Limit Comparison Method:**
- Use the limit comparison test to show convergence.
- Identify a function \( h(x) \) with known convergence properties.
- Compute \(\lim_{x \to \infty} \frac{\frac{x^2 + 3x + 7}{x^4 + 2x}}{h(x)}\) and use the result to conclude about the convergence of the original integral.
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