Answer :
To solve the problem, we need to determine the maximum value of [tex]\(\delta > 0\)[/tex] that ensures the function [tex]\(f(x) = 5 - \frac{x}{2}\)[/tex] remains within [tex]\(\varepsilon = 0.4\)[/tex] of the limit [tex]\(L = 3\)[/tex] when [tex]\(x\)[/tex] is within [tex]\(\delta\)[/tex] units of [tex]\(c = 4\)[/tex].
### Step-by-Step Solution
1. Determine the function's form:
The function given is [tex]\(f(x) = 5 - \frac{x}{2}\)[/tex].
2. Restate the limit property:
We want to find a [tex]\(\delta > 0\)[/tex] such that for all [tex]\(x\)[/tex] satisfying [tex]\(0 < |x - 4| < \delta\)[/tex] (i.e., [tex]\(x\)[/tex] is within [tex]\(\delta\)[/tex] units of 4), the following inequality holds:
[tex]\[ |(5 - \frac{x}{2}) - 3| < 0.4. \][/tex]
3. Simplify the absolute value expression:
Rewrite the expression inside the absolute value:
[tex]\[ |(5 - \frac{x}{2}) - 3| = |2 - \frac{x}{2}|. \][/tex]
4. Set up the inequality:
Now, we need to solve the inequality:
[tex]\[ |2 - \frac{x}{2}| < 0.4. \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Convert the absolute value inequality into a double inequality:
[tex]\[ -0.4 < 2 - \frac{x}{2} < 0.4. \][/tex]
6. Isolate [tex]\(x\)[/tex] on both sides:
Solve each part of the inequality separately:
- For the left side:
[tex]\[ -0.4 < 2 - \frac{x}{2} \][/tex]
[tex]\[ -2.4 < -\frac{x}{2} \][/tex]
Multiply through by -2 (note that this reverses the inequality):
[tex]\[ 4.8 > x \quad \text{or} \quad x < 4.8. \][/tex]
- For the right side:
[tex]\[ 2 - \frac{x}{2} < 0.4 \][/tex]
[tex]\[ 1.6 < \frac{x}{2} \][/tex]
Multiply through by 2:
[tex]\[ 3.2 < x \quad \text{or} \quad x > 3.2. \][/tex]
7. Combine the results:
Combining both parts, we get:
[tex]\[ 3.2 < x < 4.8. \][/tex]
8. Relate to [tex]\(\delta\)[/tex]:
We need [tex]\(0 < |x - 4| < \delta\)[/tex]. From the bounds calculated, we observe the maximum deviation from 4 is from either end of the interval [tex]\(3.2\)[/tex] and [tex]\(4.8\)[/tex]:
- From 4:
[tex]\[ 4 - 3.2 = 0.8 \][/tex]
[tex]\[ 4.8 - 4 = 0.8. \][/tex]
So, the maximum [tex]\(\delta\)[/tex] that satisfies the condition is:
[tex]\[ \boxed{0.8} \][/tex]
### Step-by-Step Solution
1. Determine the function's form:
The function given is [tex]\(f(x) = 5 - \frac{x}{2}\)[/tex].
2. Restate the limit property:
We want to find a [tex]\(\delta > 0\)[/tex] such that for all [tex]\(x\)[/tex] satisfying [tex]\(0 < |x - 4| < \delta\)[/tex] (i.e., [tex]\(x\)[/tex] is within [tex]\(\delta\)[/tex] units of 4), the following inequality holds:
[tex]\[ |(5 - \frac{x}{2}) - 3| < 0.4. \][/tex]
3. Simplify the absolute value expression:
Rewrite the expression inside the absolute value:
[tex]\[ |(5 - \frac{x}{2}) - 3| = |2 - \frac{x}{2}|. \][/tex]
4. Set up the inequality:
Now, we need to solve the inequality:
[tex]\[ |2 - \frac{x}{2}| < 0.4. \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Convert the absolute value inequality into a double inequality:
[tex]\[ -0.4 < 2 - \frac{x}{2} < 0.4. \][/tex]
6. Isolate [tex]\(x\)[/tex] on both sides:
Solve each part of the inequality separately:
- For the left side:
[tex]\[ -0.4 < 2 - \frac{x}{2} \][/tex]
[tex]\[ -2.4 < -\frac{x}{2} \][/tex]
Multiply through by -2 (note that this reverses the inequality):
[tex]\[ 4.8 > x \quad \text{or} \quad x < 4.8. \][/tex]
- For the right side:
[tex]\[ 2 - \frac{x}{2} < 0.4 \][/tex]
[tex]\[ 1.6 < \frac{x}{2} \][/tex]
Multiply through by 2:
[tex]\[ 3.2 < x \quad \text{or} \quad x > 3.2. \][/tex]
7. Combine the results:
Combining both parts, we get:
[tex]\[ 3.2 < x < 4.8. \][/tex]
8. Relate to [tex]\(\delta\)[/tex]:
We need [tex]\(0 < |x - 4| < \delta\)[/tex]. From the bounds calculated, we observe the maximum deviation from 4 is from either end of the interval [tex]\(3.2\)[/tex] and [tex]\(4.8\)[/tex]:
- From 4:
[tex]\[ 4 - 3.2 = 0.8 \][/tex]
[tex]\[ 4.8 - 4 = 0.8. \][/tex]
So, the maximum [tex]\(\delta\)[/tex] that satisfies the condition is:
[tex]\[ \boxed{0.8} \][/tex]
