https://file.aiquickdraw.com/mathbot/file/1726533259549dyltl0sl.jpg

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!

Use the given graphs of the function f (in blue) and g (in red) to find the following limits:
1. lim (f(x) + g(x)) as x approaches 1
2. lim (f(x) + g(x)) as x approaches 2
3. lim (f(x)g(x)) as x approaches 0
4. lim (f(x)/g(x)) as x approaches 4 from the left
5. lim sqrt(3 + f(x)) as x approaches 5 from the right

Find the limits for combined functions f(x) and g(x) using graphs. This problem involves limit sum, product, and quotient rules, with a square root function application.

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