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:

Estimating the Limit:

• As $x$ approaches $-2$ from below, $F(x)$ approaches $-1$.
• As $x$ approaches $-2$ from above, $F(x)$ approaches \