To find the maximum value of [tex]\(\delta > 0\)[/tex] that satisfies the limit claim for [tex]\(\varepsilon = 0.2\)[/tex], let's go through the problem step-by-step.
We are given:
[tex]\[
\lim _{x \rightarrow 2}\left(5-\frac{x}{2}\right)=4
\][/tex]
This means that as [tex]\(x\)[/tex] approaches [tex]\(2\)[/tex], the function [tex]\(f(x) = 5 - \frac{x}{2}\)[/tex] approaches the value [tex]\(4\)[/tex].
We need to find the maximum value of [tex]\(\delta\)[/tex] such that:
[tex]\[
0 < |x - 2| < \delta \implies |f(x) - 4| < 0.2
\][/tex]
1. Evaluate the inequality [tex]\(|f(x) - 4| < 0.2\)[/tex]:
[tex]\[
|(5 - \frac{x}{2}) - 4| < 0.2
\][/tex]
Simplify the expression inside the absolute value:
[tex]\[
|1 - \frac{x}{2}| < 0.2
\][/tex]
2. Remove the absolute value by considering the bounds:
[tex]\[
-0.2 < 1 - \frac{x}{2} < 0.2
\][/tex]
3. Solve the inequalities to find the bounds for [tex]\(x\)[/tex]:
For the lower bound:
[tex]\[
-0.2 < 1 - \frac{x}{2}
\][/tex]
[tex]\[
-1.2 < -\frac{x}{2}
\][/tex]
[tex]\[
1.2 > \frac{x}{2}
\][/tex]
[tex]\[
x < 2.4
\][/tex]
For the upper bound:
[tex]\[
1 - \frac{x}{2} < 0.2
\][/tex]
[tex]\[
1 - 0.2 < \frac{x}{2}
\][/tex]
[tex]\[
0.8 < \frac{x}{2}
\][/tex]
[tex]\[
1.6 < x
\][/tex]
4. Combine the bounds to get the interval for [tex]\(x\)[/tex]:
[tex]\[
1.6 < x < 2.4
\][/tex]
5. Determine [tex]\(\delta\)[/tex]:
The maximum [tex]\(\delta\)[/tex] is the smallest distance from [tex]\(x = 2\)[/tex] to either endpoint of the interval [tex]\(1.6 < x < 2.4\)[/tex].
[tex]\[
\delta_{\text{lower}} = 2 - 1.6 = 0.4
\][/tex]
[tex]\[
\delta_{\text{upper}} = 2.4 - 2 = 0.4
\][/tex]
Therefore, [tex]\(\delta = \min(\delta_{\text{lower}}, \delta_{\text{upper}}) = 0.4\)[/tex].
So, the maximum value of [tex]\(\delta\)[/tex] that satisfies the given limit claim is:
[tex]\[
\delta = 0.4
\][/tex]
