Solved on Feb 06, 2024

(a) Find the largest $\delta$ such that $|x-1|<\delta$ implies $|4x-4|<0.5$ . (b) Find the largest $\delta$ such that $|x-1|<\delta$ implies $|4x-4|<0.05$ .

STEP 1

Assumptions
1. We are given an inequality $|x - 1| < \delta$ .
2. We need to find the largest $\delta$ such that $|4x - 4| < \varepsilon$ .
3. For part (a), $\varepsilon = 0.5$ .
4. For part (b), $\varepsilon = 0.05$ .
5. The value of $\delta$ must be non-negative.

STEP 2

First, let's manipulate the inequality $|4x - 4| < \varepsilon$ to express it in terms of $|x - 1|$ .
$|4x - 4| = |4(x - 1)|$

STEP 3

Since $|ab| = |a||b|$ for any real numbers $a$ and $b$ , we can rewrite the inequality as:
$|4||x - 1| < \varepsilon$

STEP 4

Now, divide both sides of the inequality by $|4|$ to isolate $|x - 1|$ .
$|x - 1| < \frac{\varepsilon}{|4|}$

STEP 5

Since $|4| = 4$ , we can simplify the inequality to:
$|x - 1| < \frac{\varepsilon}{4}$

STEP 6

Now, we will find the largest $\delta$ for part (a) where $\varepsilon = 0.5$ .
$\delta = \frac{\varepsilon}{4}$

STEP 7

Substitute $\varepsilon = 0.5$ into the equation.
$\delta = \frac{0.5}{4}$

STEP 8

Calculate the value of $\delta$ for part (a).
$\delta = \frac{0.5}{4} = 0.125$
For part (a), the largest number $\delta$ such that if $|x - 1| < \delta$ , then $|4x - 4| < 0.5$ is $\delta \leq 0.125$ .

STEP 9

Next, we will find the largest $\delta$ for part (b) where $\varepsilon = 0.05$ .
$\delta = \frac{\varepsilon}{4}$

STEP 10

Substitute $\varepsilon = 0.05$ into the equation.
$\delta = \frac{0.05}{4}$

STEP 11

Calculate the value of $\delta$ for part (b).
$\delta = \frac{0.05}{4} = 0.0125$
For part (b), the largest number $\delta$ such that if $|x - 1| < \delta$ , then $|4x - 4| < 0.05$ is $\delta \leq 0.0125$ .

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(a) Find the largest $\delta$ such that $|x-1|<\delta$ implies $|4x-4|<0.5$ . (b) Find the largest $\delta$ such that $|x-1|<\delta$ implies $|4x-4|<0.05$ .

STEP 1

Assumptions
1. We are given an inequality $|x - 1| < \delta$ .
2. We need to find the largest $\delta$ such that $|4x - 4| < \varepsilon$ .
3. For part (a), $\varepsilon = 0.5$ .
4. For part (b), $\varepsilon = 0.05$ .
5. The value of $\delta$ must be non-negative.

STEP 2

First, let's manipulate the inequality $|4x - 4| < \varepsilon$ to express it in terms of $|x - 1|$ .
$|4x - 4| = |4(x - 1)|$

STEP 3

Since $|ab| = |a||b|$ for any real numbers $a$ and $b$ , we can rewrite the inequality as:
$|4||x - 1| < \varepsilon$

STEP 4

Now, divide both sides of the inequality by $|4|$ to isolate $|x - 1|$ .
$|x - 1| < \frac{\varepsilon}{|4|}$

STEP 5

Since $|4| = 4$ , we can simplify the inequality to:
$|x - 1| < \frac{\varepsilon}{4}$

STEP 6

Now, we will find the largest $\delta$ for part (a) where $\varepsilon = 0.5$ .
$\delta = \frac{\varepsilon}{4}$

STEP 7

Substitute $\varepsilon = 0.5$ into the equation.
$\delta = \frac{0.5}{4}$

STEP 8

Calculate the value of $\delta$ for part (a).
$\delta = \frac{0.5}{4} = 0.125$
For part (a), the largest number $\delta$ such that if $|x - 1| < \delta$ , then $|4x - 4| < 0.5$ is $\delta \leq 0.125$ .

STEP 9

Next, we will find the largest $\delta$ for part (b) where $\varepsilon = 0.05$ .
$\delta = \frac{\varepsilon}{4}$

STEP 10

Substitute $\varepsilon = 0.05$ into the equation.
$\delta = \frac{0.05}{4}$

STEP 11

Calculate the value of $\delta$ for part (b).
$\delta = \frac{0.05}{4} = 0.0125$
For part (b), the largest number $\delta$ such that if $|x - 1| < \delta$ , then $|4x - 4| < 0.05$ is $\delta \leq 0.0125$ .

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(a) Find the largest δ\deltaδ such that ∣x−1∣<δ|x-1|<\delta∣x−1∣<δ implies ∣4x−4∣<0.5|4x-4|<0.5∣4x−4∣<0.5. (b) Find the largest δ\deltaδ such that ∣x−1∣<δ|x-1|<\delta∣x−1∣<δ implies ∣4x−4∣<0.05|4x-4|<0.05∣4x−4∣<0.05.

STEP 1

STEP 2

First, let's manipulate the inequality ∣4x−4∣<ε|4x - 4| < \varepsilon∣4x−4∣<ε to express it in terms of ∣x−1∣|x - 1|∣x−1∣.∣4x−4∣=∣4(x−1)∣|4x - 4| = |4(x - 1)|∣4x−4∣=∣4(x−1)∣

STEP 3

Since ∣ab∣=∣a∣∣b∣|ab| = |a||b|∣ab∣=∣a∣∣b∣ for any real numbers aaa and bbb, we can rewrite the inequality as:∣4∣∣x−1∣<ε|4||x - 1| < \varepsilon∣4∣∣x−1∣<ε

STEP 4

Now, divide both sides of the inequality by ∣4∣|4|∣4∣ to isolate ∣x−1∣|x - 1|∣x−1∣.∣x−1∣<ε∣4∣|x - 1| < \frac{\varepsilon}{|4|}∣x−1∣<∣4∣ε​

STEP 5

Since ∣4∣=4|4| = 4∣4∣=4, we can simplify the inequality to:∣x−1∣<ε4|x - 1| < \frac{\varepsilon}{4}∣x−1∣<4ε​

STEP 6

Now, we will find the largest δ\deltaδ for part (a) where ε=0.5\varepsilon = 0.5ε=0.5.δ=ε4\delta = \frac{\varepsilon}{4}δ=4ε​

STEP 7

Substitute ε=0.5\varepsilon = 0.5ε=0.5 into the equation.δ=0.54\delta = \frac{0.5}{4}δ=40.5​

STEP 8

STEP 9

Next, we will find the largest δ\deltaδ for part (b) where ε=0.05\varepsilon = 0.05ε=0.05.δ=ε4\delta = \frac{\varepsilon}{4}δ=4ε​

STEP 10

Substitute ε=0.05\varepsilon = 0.05ε=0.05 into the equation.δ=0.054\delta = \frac{0.05}{4}δ=40.05​

STEP 11

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(a) Find the largest $\delta$ such that $|x-1|<\delta$ implies $|4x-4|<0.5$ . (b) Find the largest $\delta$ such that $|x-1|<\delta$ implies $|4x-4|<0.05$ .

First, let's manipulate the inequality $|4x - 4| < \varepsilon$ to express it in terms of $|x - 1|$ .
$|4x - 4| = |4(x - 1)|$

Since $|ab| = |a||b|$ for any real numbers $a$ and $b$ , we can rewrite the inequality as:
$|4||x - 1| < \varepsilon$

Now, divide both sides of the inequality by $|4|$ to isolate $|x - 1|$ .
$|x - 1| < \frac{\varepsilon}{|4|}$

Since $|4| = 4$ , we can simplify the inequality to:
$|x - 1| < \frac{\varepsilon}{4}$

Now, we will find the largest $\delta$ for part (a) where $\varepsilon = 0.5$ .
$\delta = \frac{\varepsilon}{4}$

Substitute $\varepsilon = 0.5$ into the equation.
$\delta = \frac{0.5}{4}$

Next, we will find the largest $\delta$ for part (b) where $\varepsilon = 0.05$ .
$\delta = \frac{\varepsilon}{4}$

Substitute $\varepsilon = 0.05$ into the equation.
$\delta = \frac{0.05}{4}$