QuestionFind $\delta$ so that if $|x-2|<\delta$ , then $|4x-8|<0.9$ .

Studdy Solution

STEP 1

Assumptions1. We are given that $|x-|<\delta$ . We want to find a $\delta$ such that $|4x-8|<\varepsilon$
3. We are given that $\varepsilon=0.9$

STEP 2

First, we need to simplify the expression $|4x-8|$ .
$|4x-8| =4|x-2|$

STEP 3

We want to find a $\delta$ such that $|x-2|<\varepsilon$ . We can express this as $|x-2|<\varepsilon$

STEP 4

To isolate $|x-2|$ , we divide both sides of the inequality by4.
$|x-2|<\frac{\varepsilon}{4}$

STEP 5

Now, we can substitute the given value of $\varepsilon$ into the inequality.
$|x-2|<\frac{0.9}{4}$

STEP 6

Calculate the value of $\frac{0.9}{4}$ .
$|x-2|<0.225$

STEP 7

So, we find that $\delta$ should be $0.225$ to satisfy the given conditions. Therefore, if $|x-2|<0.225$ , then $|4x-|<0.9$ .
The value of $\delta$ is $0.225$ .

Was this helpful?

QuestionFind $\delta$ so that if $|x-2|<\delta$ , then $|4x-8|<0.9$ .

Studdy Solution

STEP 1

Assumptions1. We are given that $|x-|<\delta$ . We want to find a $\delta$ such that $|4x-8|<\varepsilon$
3. We are given that $\varepsilon=0.9$

STEP 2

First, we need to simplify the expression $|4x-8|$ .
$|4x-8| =4|x-2|$

STEP 3

We want to find a $\delta$ such that $|x-2|<\varepsilon$ . We can express this as $|x-2|<\varepsilon$

STEP 4

To isolate $|x-2|$ , we divide both sides of the inequality by4.
$|x-2|<\frac{\varepsilon}{4}$

STEP 5

Now, we can substitute the given value of $\varepsilon$ into the inequality.
$|x-2|<\frac{0.9}{4}$

STEP 6

Calculate the value of $\frac{0.9}{4}$ .
$|x-2|<0.225$

STEP 7

So, we find that $\delta$ should be $0.225$ to satisfy the given conditions. Therefore, if $|x-2|<0.225$ , then $|4x-|<0.9$ .
The value of $\delta$ is $0.225$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind δ\deltaδ so that if ∣x−2∣<δ|x-2|<\delta∣x−2∣<δ, then ∣4x−8∣<0.9|4x-8|<0.9∣4x−8∣<0.9.

Studdy Solution

STEP 1

Assumptions1. We are given that ∣x−∣<δ|x-|<\delta∣x−∣<δ . We want to find a δ\deltaδ such that ∣4x−8∣<ε|4x-8|<\varepsilon∣4x−8∣<ε3. We are given that ε=0.9\varepsilon=0.9ε=0.9

STEP 2

First, we need to simplify the expression ∣4x−8∣|4x-8|∣4x−8∣.∣4x−8∣=4∣x−2∣|4x-8| =4|x-2|∣4x−8∣=4∣x−2∣

STEP 3

We want to find a δ\deltaδ such that ∣x−2∣<ε|x-2|<\varepsilon∣x−2∣<ε. We can express this as∣x−2∣<ε|x-2|<\varepsilon∣x−2∣<ε

STEP 4

To isolate ∣x−2∣|x-2|∣x−2∣, we divide both sides of the inequality by4.∣x−2∣<ε4|x-2|<\frac{\varepsilon}{4}∣x−2∣<4ε​

STEP 5

Now, we can substitute the given value of ε\varepsilonε into the inequality.∣x−2∣<0.94|x-2|<\frac{0.9}{4}∣x−2∣<40.9​

STEP 6

Calculate the value of 0.94\frac{0.9}{4}40.9​.∣x−2∣<0.225|x-2|<0.225∣x−2∣<0.225

STEP 7

So, we find that δ\deltaδ should be 0.2250.2250.225 to satisfy the given conditions. Therefore, if ∣x−2∣<0.225|x-2|<0.225∣x−2∣<0.225, then ∣4x−∣<0.9|4x-|<0.9∣4x−∣<0.9.The value of δ\deltaδ is 0.2250.2250.225.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind $\delta$ so that if $|x-2|<\delta$ , then $|4x-8|<0.9$ .

Assumptions1. We are given that $|x-|<\delta$ . We want to find a $\delta$ such that $|4x-8|<\varepsilon$
3. We are given that $\varepsilon=0.9$

First, we need to simplify the expression $|4x-8|$ .
$|4x-8| =4|x-2|$

We want to find a $\delta$ such that $|x-2|<\varepsilon$ . We can express this as $|x-2|<\varepsilon$

To isolate $|x-2|$ , we divide both sides of the inequality by4.
$|x-2|<\frac{\varepsilon}{4}$

Now, we can substitute the given value of $\varepsilon$ into the inequality.
$|x-2|<\frac{0.9}{4}$

Calculate the value of $\frac{0.9}{4}$ .
$|x-2|<0.225$

So, we find that $\delta$ should be $0.225$ to satisfy the given conditions. Therefore, if $|x-2|<0.225$ , then $|4x-|<0.9$ .
The value of $\delta$ is $0.225$ .