A graphing calculator is recommended.
For the limit 
lim 
x → 3 
(x3 − 3x + 5) = 23,
 illustrate this 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.)
𝜀 = 0.2	𝛿	=	

 
Incorrect: Your answer is incorrect.
𝜀 = 0.1	𝛿	=	

To solve this problem, we need to illustrate the formal definition of a limit, which uses the epsilon-delta definition. Specifically, we need to find the values of \( \delta \) that correspond to \( \varepsilon = 0.2 \) and \( \varepsilon = 0.1 \) for the given limit:

\[
\lim_{x \to 3} (x^3 - 3x + 5) = 23
\]

### Step-by-step breakdown:

1. **Restating the problem**: We know that for the limit \( \lim_{x \to 3} (x^3 - 3x + 5) = 23 \), we want to find \( \delta \) such that for all \( x \), if \( |x - 3| < \delta \), then:

\[
|f(x) - 23| < \varepsilon
\]

where \( f(x) = x^3 - 3x + 5 \), and \( \varepsilon = 0.2 \) and \( \varepsilon = 0.1 \).

2. **Rewrite the inequality**: 
For a given \( \varepsilon \), we must solve:

\[
|f(x) - 23| < \varepsilon
\]

Substitute \( f(x) = x^3 - 3x + 5 \) and simplify:

\[
|x^3 - 3x + 5 - 23| = |x^3 - 3x - 18| < \varepsilon
\]

Now we have:

\[
|x^3 - 3x - 18| < \varepsilon
\]



A graphing calculator is recommended.
For the limit 
lim 
x → 3 
(x^3 − 3x + 5) = 23,
 illustrate this 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.)
𝜀 = 0.2 𝛿 =  

Incorrect: Your answer is incorrect.
𝜀 = 0.1 𝛿 = 

Solve for the largest δ values corresponding to ε = 0.2 and ε = 0.1 in the limit lim_{x → 3} (x^3 − 3x + 5) = 23. Learn how to apply the epsilon-delta definition of limits with a step-by-step breakdown. Suitable for advanced high school or college calculus students.

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