Let's analyze the graphs and limits from the image you uploaded:

1. ( \lim_{x \to -1} [f(x) + g(x)]:**

   • From the graph of $f(x)$ (blue), at $x = -1$, $f(x)$ appears to be $f(-1) = -1$.
   • From the graph of $g(x)$ (red), at $x = -1$, $g(x)$ appears to be $g(-1) = 1$.
   • Therefore, $\lim_{x \to -1} [f(x) + g(x)] = -1 + 1 = 0$.

2. ( \lim_{x \to 2} [f(x) + g(x)]:**

   • From the graph of $f(x)$ (blue), at $x = 2$, $f(x)$ appears to be $f(2) = 0$.
   • From the graph of $g(x)$ (red), at $x = 2$, $g(x)$ appears to be $g(2) = 3$.
   • Therefore, $\lim_{x \to 2} [f(x) + g(x)] = 0 + 3 = 3$.

3. ( \lim_{x \to 0} [f(x)g(x)]:**

   • From the graph of $f(x)$ (blue), at $x = 0$, $f(x)$ appears to be $f(0) = -3$.
   • From the graph of $g(x)$ (red), at $x = 0$, $g(x)$ appears to be $g(0) = 0$.
   • Therefore, $\lim_{x \to 0} [f(x)g(x)] = -3 \times 0 = 0$.

4. ( \lim_{x \to 0} \frac{f(x)}{g(x)}:**

   • At $x = 0$, $f(0) = -3$ and $g(0) = 0$. The limit involves division by zero.
   • Since $f(0) = -3$ and $g(0) = 0$, and as $x \to 0$, the graph of $g(x)$ approaches 0 from the positive side, the limit is approaching $-\infty$.
   • Therefore, $\lim_{x \to 0} \frac{f(x)}{g(x)} = -\infty$.

5. ( \lim_{x \to -1} \left[ \sqrt{3} + f(x) \right]:**

   • We already know that $f(-1) = -1$.
   • Therefore, $\lim_{x \to -1} \left[ \sqrt{3} + f(x) \right] = \sqrt{3} + (-1) = \sqrt{3} - 1$.

Would you like further clarification on any of these steps or additional details?

Here are some relative questions to expand understanding:

1. What is the importance of left-hand and right-hand limits in limit evaluation?
2. How can you evaluate limits algebraically if the graph is not available?
3. What happens when a limit approaches a vertical asymptote?
4. How can limits at infinity help analyze end behavior of functions?
5. How do you handle indeterminate forms like $\frac{0}{0}$ when calculating limits?

Tip: Always check for continuity at a point when evaluating limits directly from a graph. This simplifies the process and avoids undefined behavior.