The following function has domain (-1,0) u (0, infinity). F(x) = (-1+ square root of 1+x)/x. Make a table of values of f(x) for x= +\- 1, +\- 0.1, +\- 0.01, and +\- 0.001. Use your table to guess a limiting value for f(x) as x approaches 0.

We are given the function:

\[
F(x) = \frac{-1 + \sqrt{1 + x}}{x}
\]

### Domain: 
The function has a domain of \((-1, 0) \cup (0, \infty)\). It is undefined at \(x = 0\) because the denominator would be zero.

### Objective: 
We need to evaluate the function for the following values of \(x\):
\[
x = -1, 1, -0.1, 0.1, -0.01, 0.01, -0.001, 0.001
\]

### Calculation:
Let's compute the values of \(F(x)\) for each value of \(x\) and organize them in a table.

I'll start by calculating the values.

Here is the table of values for \( F(x) = \frac{-1 + \sqrt{1 + x}}{x} \):

| \( x \)      | \( F(x) \)                |
|--------------|---------------------------|
| -1           | 1.0000                    |
| 1            | 0.4142                    |
| -0.1         | 0.5132                    |
| 0.1          | 0.4881                    |
| -0.01        | 0.5013                    |
| 0.01         | 0.4988                    |
| -0.001       | 0.5001                    |
| 0.001        | 0.4999                    |

### Limiting Behavior:
As \( x \) approaches 0 from both positive and negative directions, the values of \( F(x) \) appear to approach \( 0.5 \).

Thus, based on the table, the limiting value of \( F(x) \) as \( x \to 0 \) seems to be approximately **0.5**.

---

Would you like further details or have any questions about these calculations?

Here are 5 related questions for deeper understanding:
1. How can we confirm this limit mathematically?
2. Why does the function behave differently for positive and negative values of \(x\)?
3. Can we simplify \(F(x)\) near \(x = 0\) using a series expansion?
4. How does this function behave as \(x \to -1\)?
5. What would happen if we tried to define \(F(0)\)?

**Tip:** When analyzing limits numerically, always check values from both sides of the point of interest to ensure consistent behavior.

The following function has domain (-1,0) u (0, infinity). F(x) = (-1+ square root of 1+x)/x. Make a table of values of f(x) for x = +/- 1, +/- 0.1, +/- 0.01, and +/- 0.001. Use your table to guess a limiting value for f(x) as x approaches 0.

Explore how to evaluate the function F(x) = (-1 + √(1 + x)) / x for values near x = 0 and use a table of values to estimate the limit. This problem covers concepts of functions and limits, suitable for Grades 10-12.

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