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The given problem is asking to analyze the function \( F(x) = \frac{x^2 + 3x + 2}{2 - |x|} \).

### Part (a) asks to:
1. Make tables of values of \( F(x) \) as \( x \) approaches \( c = -2 \) from above (right-hand limit) and from below (left-hand limit).
2. Estimate \( \lim_{x \to -2} F(x) \).

### Step-by-step approach:

To estimate \( \lim_{x \to -2} F(x) \), we will choose values of \( x \) slightly greater than and slightly less than \(-2\). Then we'll calculate \( F(x) \) for those values.

#### Values approaching \(-2\) from above:
These values would be slightly larger than \(-2\) (e.g., \(-1.9\), \(-1.99\)).

#### Values approaching \(-2\) from below:
These values would be slightly smaller than \(-2\) (e.g., \(-2.1\), \(-2.01\)).

### Calculate \( F(x) \) at these points:

#### 1. Approaching \(-2\) from above:
- \( x = -1.9 \)
- \( x = -1.99 \)

#### 2. Approaching \(-2\) from below:
- \( x = -2.1 \)
- \( x = -2.01 \)

I will now calculate these values.Here are the values of \( F(x) \) as \( x \) approaches \(-2\):

#### Approaching \(-2\) from above:
- \( F(-1.9) \approx -0.90 \)
- \( F(-1.99) \approx -0.99 \)

#### Approaching \(-2\) from below:
- \( F(-2.1) \approx -1.10 \)
- \( F(-2.01) \approx -1.01 \)

### Estimate of the limit:
As \( x \to -2 \), it seems that \( F(x) \) is approaching a value close to \(-1\) from both sides.

Therefore, based on these calculations, we estimate:
\[
\lim_{x \to -2} F(x) \approx -1
\]

Would you like more details or further clarifications on this problem?

Here are 5 related questions for further exploration:
1. What happens to \( F(x) \) as \( x \to 0 \)?
2. Can you confirm the limit \( \lim_{x \to -2} F(x) \) algebraically?
3. How does the graph of \( F(x) \) behave near \( x = -2 \)?
4. Are there any discontinuities in \( F(x) \)? If so, where?
5. How does the absolute value in the denominator affect the behavior of \( F(x) \)?

**Tip**: When calculating limits numerically, always use values increasingly close to the point of interest to ensure precision.

Let F(x) = (x^2 + 3x + 2) / (2 - |x|). Solve part (a).

Analyze the behavior of the function F(x) = (x^2 + 3x + 2) / (2 - |x|) by calculating the limit as x approaches -2. This problem involves using limits and understanding the impact of absolute value on the function's behavior. Suitable for college calculus or advanced high school students.

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