Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To illustrate the definition given in the problem, we need to find the largest values of [tex]\(\delta\)[/tex] that correspond to the given values of [tex]\(c\)[/tex] using the limit [tex]\(\lim_{x \rightarrow 4}(2x-2)=6\)[/tex].
Given the limit [tex]\(\lim_{x \rightarrow 4}(2x-2)=6\)[/tex], our goal is to find [tex]\(\delta\)[/tex] such that:
[tex]\[ |2x - 2 - 6| < c \][/tex]
for each given value of [tex]\(c\)[/tex].
Let's tackle each [tex]\(c\)[/tex] value step by step:
### Step-by-step Solution:
#### For [tex]\(c = 0.5\)[/tex]:
We start by solving the inequality:
[tex]\[ |2x - 8| < 0.5 \][/tex]
This absolute value inequality can be split into two inequalities:
[tex]\[ -0.5 < 2x - 8 < 0.5 \][/tex]
Now solve for [tex]\(x\)[/tex]:
[tex]\[ -0.5 < 2x - 8 \rightarrow 7.5 < 2x \rightarrow \frac{7.5}{2} < x \rightarrow 3.75 < x \][/tex]
[tex]\[ 2x - 8 < 0.5 \rightarrow 2x < 8.5 \rightarrow x < \frac{8.5}{2} \rightarrow x < 4.25 \][/tex]
Combining these, we get:
[tex]\[ 3.75 < x < 4.25 \][/tex]
Next, we determine [tex]\(\delta\)[/tex] as the maximum distance from [tex]\(x = 4\)[/tex] within this interval:
[tex]\[ \delta = \min(|4 - 3.75|, |4 - 4.25|) = \min(0.25, 0.25) = 0.25 \][/tex]
Thus, for [tex]\(c = 0.5\)[/tex]:
[tex]\[ \delta \leq 0.25 \][/tex]
#### For [tex]\(c = 0.1\)[/tex]:
Now apply the same steps to the inequality:
[tex]\[ |2x - 8| < 0.1 \][/tex]
Split into two inequalities:
[tex]\[ -0.1 < 2x - 8 < 0.1 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ -0.1 < 2x - 8 \rightarrow 7.9 < 2x \rightarrow \frac{7.9}{2} < x \rightarrow 3.95 < x \][/tex]
[tex]\[ 2x - 8 < 0.1 \rightarrow 2x < 8.1 \rightarrow x < \frac{8.1}{2} \rightarrow x < 4.05 \][/tex]
Combining these, we get:
[tex]\[ 3.95 < x < 4.05 \][/tex]
Determine [tex]\(\delta\)[/tex] as the maximum distance from [tex]\(x = 4\)[/tex]:
[tex]\[ \delta = \min(|4 - 3.95|, |4 - 4.05|) = \min(0.05, 0.05) = 0.05 \][/tex]
Thus, for [tex]\(c = 0.1\)[/tex]:
[tex]\[ \delta \leq 0.05 \][/tex]
#### For [tex]\(c = 0.05\)[/tex]:
Finally, for the inequality:
[tex]\[ |2x - 8| < 0.05 \][/tex]
Split into two inequalities:
[tex]\[ -0.05 < 2x - 8 < 0.05 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ -0.05 < 2x - 8 \rightarrow 7.95 < 2x \rightarrow \frac{7.95}{2} < x \rightarrow 3.975 < x \][/tex]
[tex]\[ 2x - 8 < 0.05 \rightarrow 2x < 8.05 \rightarrow x < \frac{8.05}{2} \rightarrow x < 4.025 \][/tex]
Combining these, we get:
[tex]\[ 3.975 < x < 4.025 \][/tex]
Determine [tex]\(\delta\)[/tex] as the maximum distance from [tex]\(x = 4\)[/tex]:
[tex]\[ \delta = \min(|4 - 3.975|, |4 - 4.025|) = \min(0.025, 0.025) = 0.025 \][/tex]
Thus, for [tex]\(c = 0.05\)[/tex]:
[tex]\[ \delta \leq 0.025 \][/tex]
### Summary:
Combining all the results, we get:
[tex]\[ \begin{array}{ll} c = 0.5 & \delta \leq 0.25 \\ c = 0.1 & \delta \leq 0.05 \\ c = 0.05 & \delta \leq 0.025 \\ \end{array} \][/tex]
Given the limit [tex]\(\lim_{x \rightarrow 4}(2x-2)=6\)[/tex], our goal is to find [tex]\(\delta\)[/tex] such that:
[tex]\[ |2x - 2 - 6| < c \][/tex]
for each given value of [tex]\(c\)[/tex].
Let's tackle each [tex]\(c\)[/tex] value step by step:
### Step-by-step Solution:
#### For [tex]\(c = 0.5\)[/tex]:
We start by solving the inequality:
[tex]\[ |2x - 8| < 0.5 \][/tex]
This absolute value inequality can be split into two inequalities:
[tex]\[ -0.5 < 2x - 8 < 0.5 \][/tex]
Now solve for [tex]\(x\)[/tex]:
[tex]\[ -0.5 < 2x - 8 \rightarrow 7.5 < 2x \rightarrow \frac{7.5}{2} < x \rightarrow 3.75 < x \][/tex]
[tex]\[ 2x - 8 < 0.5 \rightarrow 2x < 8.5 \rightarrow x < \frac{8.5}{2} \rightarrow x < 4.25 \][/tex]
Combining these, we get:
[tex]\[ 3.75 < x < 4.25 \][/tex]
Next, we determine [tex]\(\delta\)[/tex] as the maximum distance from [tex]\(x = 4\)[/tex] within this interval:
[tex]\[ \delta = \min(|4 - 3.75|, |4 - 4.25|) = \min(0.25, 0.25) = 0.25 \][/tex]
Thus, for [tex]\(c = 0.5\)[/tex]:
[tex]\[ \delta \leq 0.25 \][/tex]
#### For [tex]\(c = 0.1\)[/tex]:
Now apply the same steps to the inequality:
[tex]\[ |2x - 8| < 0.1 \][/tex]
Split into two inequalities:
[tex]\[ -0.1 < 2x - 8 < 0.1 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ -0.1 < 2x - 8 \rightarrow 7.9 < 2x \rightarrow \frac{7.9}{2} < x \rightarrow 3.95 < x \][/tex]
[tex]\[ 2x - 8 < 0.1 \rightarrow 2x < 8.1 \rightarrow x < \frac{8.1}{2} \rightarrow x < 4.05 \][/tex]
Combining these, we get:
[tex]\[ 3.95 < x < 4.05 \][/tex]
Determine [tex]\(\delta\)[/tex] as the maximum distance from [tex]\(x = 4\)[/tex]:
[tex]\[ \delta = \min(|4 - 3.95|, |4 - 4.05|) = \min(0.05, 0.05) = 0.05 \][/tex]
Thus, for [tex]\(c = 0.1\)[/tex]:
[tex]\[ \delta \leq 0.05 \][/tex]
#### For [tex]\(c = 0.05\)[/tex]:
Finally, for the inequality:
[tex]\[ |2x - 8| < 0.05 \][/tex]
Split into two inequalities:
[tex]\[ -0.05 < 2x - 8 < 0.05 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ -0.05 < 2x - 8 \rightarrow 7.95 < 2x \rightarrow \frac{7.95}{2} < x \rightarrow 3.975 < x \][/tex]
[tex]\[ 2x - 8 < 0.05 \rightarrow 2x < 8.05 \rightarrow x < \frac{8.05}{2} \rightarrow x < 4.025 \][/tex]
Combining these, we get:
[tex]\[ 3.975 < x < 4.025 \][/tex]
Determine [tex]\(\delta\)[/tex] as the maximum distance from [tex]\(x = 4\)[/tex]:
[tex]\[ \delta = \min(|4 - 3.975|, |4 - 4.025|) = \min(0.025, 0.025) = 0.025 \][/tex]
Thus, for [tex]\(c = 0.05\)[/tex]:
[tex]\[ \delta \leq 0.025 \][/tex]
### Summary:
Combining all the results, we get:
[tex]\[ \begin{array}{ll} c = 0.5 & \delta \leq 0.25 \\ c = 0.1 & \delta \leq 0.05 \\ c = 0.05 & \delta \leq 0.025 \\ \end{array} \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
