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From the image, it appears that the problem is asking about the intuitive idea of limits, specifically focusing on the function \( f(x) = \frac{7.5}{x - 4} + 5 \). Here's a summary of the three parts of the question:

### Problem Breakdown:
**Part (a):** You are given a table with values for \( x \) approaching 4 (from both sides) and asked to calculate corresponding values of \( f(x) \). The table contains:
- \( x \) values: 3.9, 3.99, 3.999, 3.9999, 4.0001, 4.01, 4.1
- You need to compute \( f(x) \) for each of these values.

**Part (b):** Based on the computed table in (a), you are asked to estimate what the limit of \( f(x) \) is as \( x \) approaches 4.

**Part (c):** You are instructed to graph the function \( f(x) \) and determine if the limit you've estimated in part (b) is consistent. Then, by examining the graph, find an interval around \( x = 4 \) where the value of \( f(x) \) is within 0.01 of the limit.

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### Solutions:

#### Part (a): 
We need to compute the values of \( f(x) = \frac{7.5}{x - 4} + 5 \) for each \( x \).

I will now calculate these values.

Here are the calculated values for \( f(x) = \frac{7.5}{x - 4} + 5 \) at the specified points:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
3.9 & -70 \\
3.99 & -745 \\
3.999 & -7495 \\
3.9999 & -74995 \\
4.0001 & 75005 \\
4.01 & 755 \\
4.1 & 80 \\
\hline
\end{array}
\]

#### Part (b):
As \( x \) approaches 4, the function values get very large in magnitude (both positive and negative), which suggests that the limit does not exist in a finite sense. Instead, the function approaches \( +\infty \) as \( x \to 4^+ \) and \( -\infty \) as \( x \to 4^- \).

The limit of \( f(x) \) as \( x \) approaches 4 does not exist due to this vertical asymptote.

#### Part (c):
To graph the function and find the window around \( x = 4 \) where \( f(x) \) is within 0.01 of the limit, you would typically visualize the graph. Since we concluded that the limit approaches infinity, the graph will show a vertical asymptote at \( x = 4 \), and it's impossible for \( f(x) \) to be within 0.01 of any finite number around that point. 

Let me know if you'd like a detailed graph or more help!

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Would you like more details on any of these parts? Here are some additional related questions:

1. What is a vertical asymptote, and how does it affect the behavior of a function?
2. Can a limit be infinite? If so, what does that mean?
3. How can you determine if a limit does not exist?
4. What is the formal definition of a limit?
5. How do you graph functions with asymptotes?

**Tip:** When a function has a vertical asymptote, you can always check whether the limit approaches positive or negative infinity by looking at the behavior on either side of the asymptote.

Homework 04 -- Intuitive Idea of Limits: Problem 2. The function f(x) = (7.5 / (x - 4)) + 5 is given. (a) Calculate the values of f(x) as x approaches 4 from both sides. (b) Estimate the limit of f(x) as x approaches 4 based on the calculated values. (c) Graph the function and determine an interval near x = 4 where f(x) is within 0.01 of the limit.

Explore the intuitive idea of limits with the function f(x) = (7.5 / (x - 4)) + 5. Solve for values of f(x) as x approaches 4 and analyze the limit, including graphing and asymptotic behavior. Suitable for pre-calculus students.

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