Answer :
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]
