At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To solve this problem, we need to identify the maximum value of [tex]\(\delta > 0\)[/tex] such that the condition [tex]\(0 < |x - c| < \delta\)[/tex] assures [tex]\(f(x) > 200\)[/tex], given that [tex]\( f(x) = \frac{3}{x^2} \)[/tex] and [tex]\( \lim _{x \rightarrow 0} \frac{3}{x^2} = \infty \)[/tex].
### Step-by-Step Solution:
1. Understand the Function Behavior Near Zero:
The function [tex]\( f(x) = \frac{3}{x^2} \)[/tex] tends towards infinity as [tex]\(x\)[/tex] approaches 0.
2. Set Up the Inequality:
We are given that [tex]\( f(x) > 200 \)[/tex]. Thus, we need to solve for [tex]\(x\)[/tex] in the inequality:
[tex]\[ \frac{3}{x^2} > 200 \][/tex]
3. Solve the Inequality:
To isolate [tex]\(|x|\)[/tex], we proceed as follows:
[tex]\[ \frac{3}{x^2} > 200 \][/tex]
[tex]\[ x^2 < \frac{3}{200} \][/tex]
[tex]\[ x^2 < 0.015 \][/tex]
4. Take the Square Root:
To find the constraint on [tex]\(|x|\)[/tex], take the square root of both sides:
[tex]\[ |x| < \sqrt{0.015} \][/tex]
5. Compute the Square Root:
Calculate [tex]\(\sqrt{0.015}\)[/tex]:
[tex]\[ \sqrt{0.015} \approx 0.1224744871 \][/tex]
6. Round Down to Two Decimal Places:
To get the maximum value of [tex]\(\delta\)[/tex], we round down to two decimal places:
[tex]\[ \delta \approx 0.12 \][/tex]
Thus, the maximum value of [tex]\(\delta\)[/tex] that satisfies the condition is:
[tex]\[ \delta = 0.12 \][/tex]
### Step-by-Step Solution:
1. Understand the Function Behavior Near Zero:
The function [tex]\( f(x) = \frac{3}{x^2} \)[/tex] tends towards infinity as [tex]\(x\)[/tex] approaches 0.
2. Set Up the Inequality:
We are given that [tex]\( f(x) > 200 \)[/tex]. Thus, we need to solve for [tex]\(x\)[/tex] in the inequality:
[tex]\[ \frac{3}{x^2} > 200 \][/tex]
3. Solve the Inequality:
To isolate [tex]\(|x|\)[/tex], we proceed as follows:
[tex]\[ \frac{3}{x^2} > 200 \][/tex]
[tex]\[ x^2 < \frac{3}{200} \][/tex]
[tex]\[ x^2 < 0.015 \][/tex]
4. Take the Square Root:
To find the constraint on [tex]\(|x|\)[/tex], take the square root of both sides:
[tex]\[ |x| < \sqrt{0.015} \][/tex]
5. Compute the Square Root:
Calculate [tex]\(\sqrt{0.015}\)[/tex]:
[tex]\[ \sqrt{0.015} \approx 0.1224744871 \][/tex]
6. Round Down to Two Decimal Places:
To get the maximum value of [tex]\(\delta\)[/tex], we round down to two decimal places:
[tex]\[ \delta \approx 0.12 \][/tex]
Thus, the maximum value of [tex]\(\delta\)[/tex] that satisfies the condition is:
[tex]\[ \delta = 0.12 \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.