Looking at the image, the problem asks to determine limits based on the graphs of the functions $f(x)$ (in blue) and $g(x)$ (in red). The graph has certain key points at which limits need to be evaluated.

Let's go through the limits requested:

1. $\lim_{x \to 1} (f(x) + g(x))$:

   • We need the values of $f(x)$ and $g(x)$ as $x$ approaches 1. From the graph:
     • $f(x)$ as $x \to 1$ seems to approach 0.
     • $g(x)$ as $x \to 1$ approaches 1.
   • Therefore, $\lim_{x \to 1} (f(x) + g(x)) = 0 + 1 = 1$.

2. $\lim_{x \to 2} (f(x) + g(x))$:

   • Check the values of $f(x)$ and $g(x)$ as $x$ approaches 2.
     • $f(x) \to -2$ as $x \to 2$ from the graph.
     • $g(x) \to 2$ as $x \to 2$.
   • Hence, $\lim_{x \to 2} (f(x) + g(x)) = -2 + 2 = 0$.

3. $\lim_{x \to 0} f(x)g(x)$:

   • As $x \to 0$, the values are:
     • $f(x) \to -1$ and $g(x) \to 0$.
   • Therefore, $\lim_{x \to 0} f(x)g(x) = (-1) \times 0 = 0$.

4. $\lim_{x \to 4^-} \frac{f(x)}{g(x)}$:

   • From the left-hand side of $x = 4$:
     • $f(x) \to 2$ as $x \to 4^-$.
     • $g(x) \to 2$ as $x \to 4^-$.
   • Therefore, $\lim_{x \to 4^-} \frac{f(x)}{g(x)} = \frac{2}{2} = 1$.

5. $\lim_{x \to 5^+} \sqrt{3 + f(x)}$:

   • From the right-hand side of $x = 5$:
     • $f(x) \to 6$ as $x \to 5^+$.
   • Thus, $\lim_{x \to 5^+} \sqrt{3 + f(x)} = \sqrt{3 + 6} = \sqrt{9} = 3$.

Let me know if you'd like more details, or if anything needs clarification.

Here are 5 related questions to deepen your understanding:

1. How do one-sided limits affect overall limit behavior at discontinuities?
2. What happens if one or both functions in a limit expression are discontinuous at the point of interest?
3. Can you determine the limit if a function approaches infinity? What would that mean for $f(x) + g(x)$?
4. How would the limit change if the function had an oscillating behavior near the point?
5. What role does the square root in question 5 play when dealing with limits?

Tip: Always check both the left-hand and right-hand limits when evaluating discontinuities!