A graphing calculator is recommended.
For the limit 
lim 
x → 2 
(x3 − 5x + 2) = 0,
 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.)

We are tasked with illustrating the definition of a limit using the limit:

\[
\lim_{x \to 2} (x^3 - 5x + 2) = 0
\]

The goal is to find the largest possible values of \( \delta \) corresponding to \( \varepsilon = 0.2 \) and \( \varepsilon = 0.1 \).

### General Idea
According to the formal definition of a limit, we need to find \( \delta > 0 \) such that:

\[
|x - 2| < \delta \quad \text{implies} \quad |(x^3 - 5x + 2) - 0| < \varepsilon
\]

This means that the values of the function \( f(x) = x^3 - 5x + 2 \) must be within \( \varepsilon \) of 0 when \( x \) is within \( \delta \) of 2.

### Steps

1. **Determine the function:**
   The function in question is:
   \[
   f(x) = x^3 - 5x + 2
   \]
   We want to find \( \delta \) such that:
   \[
   |x^3 - 5x + 2| < \varepsilon
   \]

2. **Evaluate \( f(2) \):**
   To check that the limit approaches 0:
   \[
   f(2) = 2^3 - 5(2) + 2 = 8 - 10 + 2 = 0
   \]
   So, \( f(2) = 0 \), confirming that \( \lim_{x \to 2} f(x) = 0 \).

3. **Solve for \( \delta \):**

   We need to find the largest \( \delta \) such that:
   \[
   |(x^3 - 5x + 2)| < \varepsilon
   \]
   for \( \varepsilon = 0.2 \) and \( \varepsilon = 0.1 \). We will solve numerically for \( \delta \) by finding when the absolute difference between the function value and 0 equals 0.2 and 0.1.

Let me calculate this step.It seems there was an issue with directly solving the equation involving the absolute value. To resolve this, I'll compute the values of \( f(x) = x^3 - 5x + 2 \) around \( x = 2 \) numerically and find the appropriate \( \delta \) values for the given \( \varepsilon = 0.2 \) and \( \varepsilon = 0.1 \).

Let me proceed by calculating \( \delta \) numerically.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the steps to solve this problem manually.

Would you like further clarification on this process? 

Meanwhile, here are 5 related questions to explore further:
1. How is the definition of a limit formally stated?
2. How can the delta-epsilon definition of limits be applied in other cases?
3. What role does continuity play in limits?
4. What is the difference between a one-sided and two-sided limit?
5. How can graphing calculators assist in visualizing limits?

**Tip**: When evaluating limits numerically, try graphing the function to visually estimate how the function behaves around the target point.

A graphing calculator is recommended. For the limit lim x → 2 (x^3 − 5x + 2) = 0, 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.)

Explore how to use the delta-epsilon definition to find the largest δ values corresponding to ε = 0.2 and ε = 0.1 for the limit lim x → 2 (x^3 − 5x + 2) = 0. This calculus problem involves understanding limits and how to apply the formal limit definition.

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