For the limit
lim
x → 2
(x3 − 5x + 3) = 1
illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1. (Round your answers to four decimal places.)
i set it up like this
1-0.2</x^3-5x+3/<0.2+1
0.8</x^3-5x+3/<1.2
then I replaced 0.8 and 1.2 in tho the function and f(0.8)= -0.488
f(1.2)=-1.272
do i add and subtract these from 2?
For ε=0.1, you want to find δ such that
|f(2+δ)-f(2)| < 0.1
|((2+δ)^3-5(2+δ)+3)-(2^3-5*2+3)| < 0.1
|δ^3+6δ^2+7δ| < 0.1
δ < 0.014
Well, you're on the right track! To find the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1, you want to select the smallest value of δ that satisfies the inequality:
|x^3 - 5x + 3 - 1| < ε
For ε = 0.2:
0.8 < x^3 - 5x + 3 < 1.2
Then, you replaced 0.8 and 1.2 into the function, which is a good first step. However, you don't need to add or subtract these values from 2 just yet. Instead, you should solve for x to find the corresponding values.
For ε = 0.2, you found that f(0.8) = -0.488 and f(1.2) = -1.272. These values indicate the range of the function for your chosen interval. To compute the respective values of x, you'll need to set up two equations:
For f(0.8):
-0.2 < 0.8^3 - 5(0.8) + 3 - 1 < 0.2
For f(1.2):
-0.2 < 1.2^3 - 5(1.2) + 3 - 1 < 0.2
Solving these equations will give you the values of x that correspond to ε = 0.2. Similarly, you can repeat this process for ε = 0.1 to find the corresponding values of x.
Remember, the goal is to find the smallest values of δ that satisfy the inequalities. I hope this clarifies the next steps for you!
To illustrate the definition of the limit, you can add and subtract the values you obtained from evaluating the function at ε = 0.2 and ε = 0.1 from the given value of x, which is 2.
For ε = 0.2:
- Adding -0.488 and subtracting 2 would give you a lower bound of (2 - 0.488) = 1.512.
- Adding -1.272 and subtracting 2 would give you an upper bound of (2 - 1.272) = 0.728.
So, for ε = 0.2, the largest possible values of δ that correspond to this value of ε would be between 0.728 and 1.512.
For ε = 0.1:
- Adding -0.488 and subtracting 2 would give you a lower bound of (2 - 0.488) = 1.512.
- Adding -1.272 and subtracting 2 would give you an upper bound of (2 - 1.272) = 0.728.
So, for ε = 0.1, the largest possible values of δ that correspond to this value of ε would also be between 0.728 and 1.512.
To find the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1, you can follow these steps:
Step 1: Start with the inequality |f(x) - 1| < ε, where f(x) = x^3 - 5x + 3 and ε is the given value, either 0.2 or 0.1.
Step 2: Rewrite the inequality as -ε < f(x) - 1 < ε.
Step 3: Replace f(x) with its expression: -ε < x^3 - 5x + 3 - 1 < ε.
Step 4: Simplify: -ε < x^3 - 5x + 2 < ε.
Now, you correctly set up the inequality for the function as 0.8 < x^3 - 5x + 2 < 1.2, using ε = 0.2.
To find the largest possible values of δ, you need to find the values of x that satisfy this inequality.
Step 5: Evaluate the function at the lower and upper bounds.
f(0.8) = (0.8)^3 - 5(0.8) + 3 = 0.928
f(1.2) = (1.2)^3 - 5(1.2) + 3 = 1.568
Step 6: Determine the range of values you need to consider around x = 2 by subtracting and adding these values to 2.
2 - 0.928 ≈ 1.072
2 + 1.568 ≈ 3.568
So, values of x between approximately 1.072 and 3.568 will satisfy the inequality |f(x) - 1| < ε, where ε = 0.2.
You can follow the same steps to find the largest possible values of δ for ε = 0.1 by substituting ε = 0.1 in step 4.