The function given is:

$F(x) = \frac{x^2 + 3x + 2}{2 - |x|}$

We are asked to estimate the limit as $x$ approaches $-2$ from both sides, i.e., $\lim_{x \to -2} F(x)$.

Step-by-Step Process:

1. Analyze the function: The absolute value function $|x|$ causes two cases:

   • When $x \geq 0$, $|x| = x$.
   • When $x < 0$, $|x| = -x$.

2. Case when $x \geq 0$: For $x \geq 0$, the function becomes: $F(x) = \frac{x^2 + 3x + 2}{2 - x}$

3. Case when $x < 0$: For $x < 0$, the absolute value changes and the function becomes: $F(x) = \frac{x^2 + 3x + 2}{2 + x}$

Create Tables of Values:

We now need to create a table of values for $F(x)$ as $x$ approaches $-2$ from above and below.

Approaching from the left ( $x \to -2^-$):

Use the function $F(x) = \frac{x^2 + 3x + 2}{2 + x}$ for values of $x$ slightly less than $-2$.

Approaching from the right ( $x \to -2^+$):

Use the function $F(x) = \frac{x^2 + 3x + 2}{2 - x}$ for values of $x$ slightly more than $-2$.

Let me compute these values next.Here are the computed values for $F(x)$ as $x$ approaches $-2$:

Approaching from the left ($x \to -2^-$):

For $F(x) = \frac{x^2 + 3x + 2}{2 + x}$:

• $x = -2.1$, $F(x) = -1.1$
• $x = -2.01$, $F(x) = -1.01$
• $x = -2.001$, $F(x) = -1.001$
• $x = -2.0001$, $F(x) = -1.0001$

Approaching from the right ($x \to -2^+$):

For $F(x) = \frac{x^2 + 3x + 2}{2 - x}$:

• $x = -1.9$, $F(x) = -0.02308$
• $x = -1.99$, $F(x) = -0.00248$
• $x = -1.999$, $F(x) = -0.0002498$
• $x = -1.9999$, $F(x) = -0.000024998$

Conclusion:

As $x \to -2^-$, the values of $F(x)$ approach $-1$.
As $x \to -2^+$, the values of $F(x)$ approach 0.

Thus, the left-hand limit is $-1$ and the right-hand limit is $0$, so the limit $\lim_{x \to -2} F(x)$ does not exist because the left and right limits are not equal.

Would you like further clarification or more details?

Here are 5 related questions to explore further:

1. What is the significance of left-hand and right-hand limits in determining continuity?
2. Can you explain what happens when limits from both sides are different?
3. How does the presence of absolute value functions impact the behavior of limits?
4. How would you compute the limits for values close to other points of interest in a similar function?
5. What is the definition of a removable discontinuity, and does this function have one?

Tip: Always check limits from both sides when dealing with absolute value functions or piecewise functions!