Q. 68 The Limit Comparison Test: If tw... [FREE SOLUTION]

91影视

Chapter 5: Q. 68 (page 479)

The Limit Comparison Test: If two functions f(x) and g(x) behave in a similar way on 摆补,鈭�), then their improper integrals on 摆补,鈭�) should converge or diverge together. More specifically, the limit comparison test says that if $\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)} = L$ for some positive real number L, then the improper integrals ${鈭�}_{a}^{鈭�} f (x) d x and {鈭�}_{a}^{鈭�} g (x) d x$ either both converge or both diverge. We will learn more about limit comparison tests in Chapter 7. Use this test to determine the convergence or divergence of the integrals in Exercises 65鈥�70.
${鈭�}_{1}^{鈭�} \frac{x^{2} - x + 3}{x^{4} + 1} d x$

Short Answer

Expert verified

The given integral is diverges.

Step by step solution

Step 1. Given Information.

The given integral is ${鈭�}_{1}^{鈭�} \frac{x^{2} - x + 3}{x^{4} + 1} d x .$

Step 2. Determining the integral is convergence or divergence.

To determine the convergence or divergence of the integral, we will use the limit comparison test.

Let two functionsf(x)andg(x)as: $f (x) = \frac{x^{2} - x + 3}{x^{4} + 1} and g (x) = \frac{1}{x} .$

Step 3. Estimate the limits of the functions.

So,

$\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)} = \lim_{x 鈫� 鈭�} \frac{(\frac{x^{2} - x + 3}{x^{4} + 1})}{(\frac{1}{x})} = \lim_{x 鈫� 鈭�} \frac{x^{2} - x + 3}{x (x^{4} + 1)} = \lim_{x 鈫� 鈭�} \frac{x^{2}}{x (x^{4} + 1)} - \lim_{x 鈫� 鈭�} \frac{x}{x (x^{4} + 1)} + \lim_{x 鈫� 鈭�} \frac{3}{x (x^{4} + 1)} = \lim_{x 鈫� 鈭�} \frac{1}{x^{4} (1 + \frac{1}{x^{4}})} - \lim_{x 鈫� 鈭�} \frac{1}{(x^{4} + 1)} + \lim_{x 鈫� 鈭�} \frac{3}{x (x^{4} + 1)} = 0 [\frac{1}{鈭�} = 0]$

Since 0 is a nonnegative real number. Thus, by the limit comparison test, both the improper integrals ${鈭�}_{1}^{鈭�} \frac{1}{x} d x and {鈭�}_{1}^{鈭�} \frac{x^{2} - x + 3}{x^{4} + 1} dx$ either both converge or diverge.

By improper integral for power functions ${鈭�}_{1}^{鈭�} \frac{1}{x} d x$ diverges.

Thus, the improper integral ${鈭�}_{1}^{鈭�} \frac{x^{2} - x + 3}{x^{4} + 1} d x$ also diverges.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Determining the integral is convergence or divergence.

Step 3. Estimate the limits of the functions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Mechanics Maths

Calculus

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.