Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
