A graphing calculator is recommended.
For the limit 
lim 
x → 2 
(x3 − 5x + 9) = 7,
 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	=	

 

𝜀 = 0.1	𝛿	=	

 


To solve this problem, we use the formal definition of the limit, which states that for a function \( f(x) \), \(\lim_{x \to a} f(x) = L\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \( 0 < |x - a| < \delta \), it follows that \( |f(x) - L| < \epsilon \).

Given:

\[
\lim_{x \to 2} (x^3 - 5x + 9) = 7
\]

We need to find the largest \(\delta\) corresponding to \(\epsilon = 0.2\) and \(\epsilon = 0.1\).

### Step 1: Define the function and limits
Let:

\[
f(x) = x^3 - 5x + 9
\]

and the limit value \( L = 7 \).

We want to find \(\delta\) such that:

\[
|f(x) - 7| < \epsilon
\]

for \(\epsilon = 0.2\) and \(\epsilon = 0.1\).

### Step 2: Set up the inequality for \(\epsilon\)
1. For \(\epsilon = 0.2\):

   \[
   |f(x) - 7| = |x^3 - 5x + 9 - 7| = |x^3 - 5x + 2| < 0.2
   \]

2. For \(\epsilon = 0.1\):

   \[
   |f(x) - 7| = |x^3 - 5x + 2| < 0.1
   \]

### Step 3: Solve for \(\delta\)
We need to solve these inequalities numerically to find the corresponding \(\delta\) values. I will calculate these values now.It looks like the approach to solving for \(\delta\) led to some unexpected behavior due to how the inequalities were handled. I'll refine the approach to find the correct \(\delta\) values by numerically approximating the bounds that satisfy the inequalities for each \(\epsilon\). Let me correct this and find the correct \(\delta\) values.The error indicates that the function values at the endpoints of the interval did not have opposite signs, which is required for the bisection method to work properly. To resolve this, I'll adjust the approach to correctly identify a valid range for \(\delta\) by checking smaller intervals. Let's try again with refined intervals.It seems that the interval needs to be further refined to find the correct \(\delta\) values. I'll adjust the interval and test the function's values more carefully to ensure that the signs differ within a suitable range, which will help identify the largest \(\delta\). Let’s try refining this approach further.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can still guide you through the steps you need to follow on a graphing calculator or software like Desmos to find the largest \(\delta\) values corresponding to \(\epsilon = 0.2\) and \(\epsilon = 0.1\). Would you like me to explain the steps in detail? 

Here are some related questions to expand your understanding:

1. What is the formal definition of a limit, and how is it applied in calculus?
2. How do you use a graphing calculator to find limits numerically?
3. What are the steps to graph \(f(x) = x^3 - 5x + 2\) and visually determine \(\delta\)?
4. How does changing \(\epsilon\) affect the corresponding \(\delta\) in limits?
5. What are some common challenges when finding \(\delta\) values for different \(\epsilon\)?

**Tip:** When calculating \(\delta\) values for a given \(\epsilon\), visualize the function around the target point and adjust the range until the inequality \( |f(x) - L| < \epsilon \) is satisfied.

A graphing calculator is recommended. For the limit lim x → 2 (x^3 − 5x + 9) = 7, 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 formal definition of limits to find the largest possible δ values for ε = 0.2 and ε = 0.1 in the limit lim x → 2 (x^3 - 5x + 9) = 7. This detailed step-by-step guide covers solving inequalities and finding corresponding δ values, suitable for calculus students.

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