Given that
lim (4x − 2) = 2
x→1
illustrate Definition 2 by finding values of δ that correspond to ε = 0.5, ε = 0.1, and ε = 0.05.
ε = 0.5 δ ≤
ε = 0.1 δ ≤
ε = 0.05 δ ≤
Well, well, well, it seems we've stumbled upon some limits! Let's get this clown show started!
According to Definition 2, when we talk about a limit, we say that the limit of a function f(x) as x approaches a certain value, let's say x = c, is equal to a number L if we can make the values of f(x) as close as we want to L by taking x to be sufficiently close, but not equal, to c.
So, let's find some values of δ (which represents the distance between x and c) that correspond to different epsilon (ε) values.
ε = 0.5:
We know that the limit of (4x - 2) as x approaches 1 is 2. In simple terms, this means that as x gets closer to 1, our function (4x - 2) gets closer to 2. So, let's find a value of δ (the clown distance) that corresponds to an epsilon (ε) of 0.5.
If the absolute value of (4x - 2 - 2) (which is just the difference between our function and the limit) is less than 0.5, then we have ourselves a match!
|4x - 2 - 2| < 0.5
|4x - 4| < 0.5
To find a maximum value for x within this inequality, we clown around with some algebraic tricks:
-0.5 < 4x - 4 < 0.5
3.5 < 4x < 4.5
0.875 < x < 1.125
So, any value of x within this clownish range will give us a value of (4x - 2) within a distance of 0.5 from the limit of 2. Therefore, for ε = 0.5, δ ≤ 0.125.
Now let's juggle another value!
ε = 0.1:
Using the same method, we'll find the clownish range of x values that correspond to an epsilon (ε) of 0.1.
|4x - 2 - 2| < 0.1
|4x - 4| < 0.1
Again, we clown around with algebra:
-0.1 < 4x - 4 < 0.1
3.9 < 4x < 4.1
0.975 < x < 1.025
So, for ε = 0.1, δ ≤ 0.025.
Alright, last one, I promise!
ε = 0.05:
For this final act, we'll find the clownish range of x values that correspond to an epsilon (ε) of 0.05.
|4x - 2 - 2| < 0.05
|4x - 4| < 0.05
Algebra, here we go again:
-0.05 < 4x - 4 < 0.05
3.95 < 4x < 4.05
0.9875 < x < 1.0125
So, for ε = 0.05, δ ≤ 0.0125.
And there you have it, my friend! For ε = 0.5, δ ≤ 0.125. For ε = 0.1, δ ≤ 0.025. And for ε = 0.05, δ ≤ 0.0125. Don't worry, I'll be here juggling clowns and math problems whenever you need me!
To illustrate Definition 2, we need to find values of δ that correspond to different epsilon (ε) values:
1. For ε = 0.5:
According to the definition, for any ε > 0, there exists a δ > 0 such that if 0 < |x - 1| < δ, then |(4x - 2) - 2| < ε.
Let's solve the inequality:
|4x - 2 - 2| < 0.5
|4x - 4| < 0.5
4|x - 1| < 0.5
Since we have |x - 1| < δ, we can write the inequality as follows:
4δ < 0.5
Dividing both sides of the inequality by 4, we get:
δ < 0.125
Therefore, for ε = 0.5, any δ value less than 0.125 satisfies the definition.
So, ε = 0.5 δ ≤ 0.125
2. For ε = 0.1:
Following the same steps, we have:
|4x - 4| < 0.1
4|x - 1| < 0.1
4δ < 0.1
δ < 0.025
Hence, for ε = 0.1, any δ value less than 0.025 satisfies the definition.
So, ε = 0.1 δ ≤ 0.025
3. For ε = 0.05:
Similarly, we have:
|4x - 4| < 0.05
4|x - 1| < 0.05
4δ < 0.05
δ < 0.0125
For ε = 0.05, any δ value less than 0.0125 satisfies the definition.
So, ε = 0.05 δ ≤ 0.0125
Therefore, we have determined the values of δ that correspond to ε = 0.5, ε = 0.1, and ε = 0.05:
ε = 0.5 δ ≤ 0.125
ε = 0.1 δ ≤ 0.025
ε = 0.05 δ ≤ 0.0125
To illustrate Definition 2 for the given limit, we need to find values of δ that correspond to different values of ε. Definition 2 states that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 1| < δ, then |(4x - 2) - 2| < ε.
To find the values of δ for different values of ε, we will solve the inequality |(4x - 2) - 2| < ε.
1. For ε = 0.5:
We want to find δ such that if 0 < |x - 1| < δ, then |4x - 4| < 0.5.
Let's simplify the expression: |4x - 4| < 0.5
This means -0.5 < 4x - 4 < 0.5.
Adding 4 to each part of the inequality, we get -0.5 + 4 < 4x < 0.5 + 4.
Simplifying further, 3.5 < 4x < 4.5.
Now, dividing each part by 4, we have 3.5/4 < x < 4.5/4.
This simplifies to 0.875 < x < 1.125.
Therefore, to satisfy the condition, if 0 < |x - 1| < 0.125, then |(4x - 2) - 2| < 0.5.
So, for ε = 0.5, δ ≤ 0.125.
2. For ε = 0.1:
We want to find δ such that if 0 < |x - 1| < δ, then |4x - 4| < 0.1.
Simplifying the expression, -0.1 < 4x - 4 < 0.1.
Adding 4 to each part of the inequality, we get -0.1 + 4 < 4x < 0.1 + 4.
Simplifying further, 3.9 < 4x < 4.1.
Dividing each part by 4, we have 3.9/4 < x < 4.1/4.
This simplifies to 0.975 < x < 1.025.
Therefore, for ε = 0.1, δ ≤ 0.025.
3. For ε = 0.05:
We want to find δ such that if 0 < |x - 1| < δ, then |4x - 4| < 0.05.
Simplifying the expression, -0.05 < 4x - 4 < 0.05.
Adding 4 to each part of the inequality, we get -0.05 + 4 < 4x < 0.05 + 4.
Simplifying further, 3.95 < 4x < 4.05.
Dividing each part by 4, we have 3.95/4 < x < 4.05/4.
This simplifies to 0.9875 < x < 1.0125.
Therefore, for ε = 0.05, δ ≤ 0.0125.
In summary:
- For ε = 0.5, δ ≤ 0.125.
- For ε = 0.1, δ ≤ 0.025.
- For ε = 0.05, δ ≤ 0.0125.