Chapter 7: Q.1 b) (page 630)

consider the statement if $0 \leq f (x) \leq g (x)$ for every x>0 and $\lim_{x \to \infty} \frac{f (x)}{g (x)} = 3$ , then the improper integrals $\int_{0}^{\infty} g (x) d x a n d \int_{0}^{\infty} f (x) d x$ both converge. The objective is to determine whether statement is true or false.

Short Answer

Expert verified

False

Step by step solution

To determine the statement converge or not

Consider the function $g (x) = \frac{1}{x^{2}} a n d f (x) = \frac{3}{x^{2}}$ .

The value of $\lim_{x \to \infty} \frac{f (x)}{g (x)}$ is:

$\lim_{x \to \infty} \frac{f (x)}{g (x)} = \lim_{x \to \infty} 3$

= 3

Therefore, both the integrals, $\int_{0}^{\infty} g (x) d x = \int_{0}^{\infty} \frac{1}{x^{2}} d x$ and $\int_{0}^{\infty} f (x) d x = \int_{0}^{\infty} \frac{3}{x^{2}}$ diverges.

Therefore the given statement is false.

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consider the statement if $0 \leq f (x) \leq g (x)$ for every x>0 and $\lim_{x \to \infty} \frac{f (x)}{g (x)} = 3$ , then the improper integrals $\int_{0}^{\infty} g (x) d x a n d \int_{0}^{\infty} f (x) d x$ both converge. The objective is to determine whether statement is true or false.

Short Answer

Step by step solution

To determine the statement converge or not

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consider the statement if 0≤f(x)≤g(x)for every x>0 andlimx→∞f(x)g(x)=3, then the improper integrals ∫0∞g(x)dxand∫0∞f(x)dxboth converge. The objective is to determine whether statement is true or false.

Short Answer

Step by step solution

To determine the statement converge or not

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

consider the statement if $0 \leq f (x) \leq g (x)$ for every x>0 and $\lim_{x \to \infty} \frac{f (x)}{g (x)} = 3$ , then the improper integrals $\int_{0}^{\infty} g (x) d x a n d \int_{0}^{\infty} f (x) d x$ both converge. The objective is to determine whether statement is true or false.