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.

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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.