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The problem defines the function \( F(x) = \frac{x^2 + 3x + 2}{2 - |x|} \), and asks to:

1. **Make a table of values** for \( F(x) \) as \( x \) approaches \( -2 \) from both above and below.
2. **Estimate the limit** \( \lim_{x \to -2} F(x) \).

Let's first evaluate \( F(x) \) as \( x \) approaches \( -2 \) from the left (below) and right (above). This will help estimate the behavior of the function around \( x = -2 \).

The function involves \( |x| \), meaning we'll handle values for \( |x| \) differently based on whether \( x \) is positive or negative. For \( x > 0 \), \( |x| = x \), and for \( x < 0 \), \( |x| = -x \).

### Table of Values

We will calculate the values of \( F(x) \) at points slightly above and below \( -2 \), such as:

- **From below**: \( x = -2.1, -2.01, -2.001 \)
- **From above**: \( x = -1.9, -1.99, -1.999 \)

Let me compute those values for you.Here is the table of values for \( F(x) = \frac{x^2 + 3x + 2}{2 - |x|} \) as \( x \) approaches \( -2 \) from both below and above:

| \( x \) (from below) | \( F(x) \) (below) | \( x \) (from above) | \( F(x) \) (above) |
|----------------------|---------------------|----------------------|---------------------|
| -2.1                 | -1.10               | -1.9                 | -0.90               |
| -2.01                | -1.01               | -1.99                | -0.99               |
| -2.001               | -1.001              | -1.999               | -0.999              |

### Estimating the Limit:
- As \( x \) approaches \( -2 \) from **below**, \( F(x) \) approaches \(-1\).
- As \( x \) approaches \( -2 \) from **above**, \( F(x) \) approaches \

Let F(x) = (x^2 + 3x + 2) / (2 - |x|). Make tables of values of F at values of x that approach c = -2 from above and below. Then estimate lim F(x) as x approaches -2.

This problem involves finding the limit of a rational function F(x) = (x^2 + 3x + 2) / (2 - |x|) as x approaches -2. A table of values is created for x approaching from above and below, with key algebraic concepts such as limits and absolute value functions.

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