Find the maximum value of [tex]\(\delta \ \textgreater \ 0\)[/tex] that satisfies the limit claim for [tex]\(M=200\)[/tex], such that [tex]\(0 \ \textless \ |x - c| \ \textless \ \delta\)[/tex] implies [tex]\(f(x) \ \textgreater \ 200\)[/tex]. Round down to two decimal places if necessary.

[tex]\[
\lim _{x \rightarrow 0} \frac{3}{x^2} = \infty
\][/tex]



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]