https://file.aiquickdraw.com/mathbot/file/17268131226191pwgbhdh.jpg

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.

Use the given graphs of the function f (in blue) and g (in red) to find the following limits:
1. lim[x -> -1] [f(x) + g(x)]
2. lim[x -> 2] [f(x) + g(x)]
3. lim[x -> 0] [f(x)g(x)]
4. lim[x -> 0] f(x)/g(x)
5. lim[x -> -1] [sqrt(3) + f(x)]

Learn how to find limits from given graphs of functions f(x) and g(x). This guide covers the calculation of sum, product, and quotient limits, including handling indeterminate forms and approaching infinity. Suitable for advanced high school students in Grades 11-12.

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