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To solve the given limits using the graphs of \( f(x) \) and \( g(x) \), we'll examine the behavior of the functions as \( x \) approaches 2.

1. **Limit 1:** \(\lim\limits_{x \to 2} \frac{f(x)}{g(x)}\)

   - **Behavior of \( f(x) \) as \( x \) approaches 2:**
     - From the graph of \( f(x) \), as \( x \) approaches 2 from both sides, the value of \( f(x) \) approaches 1.

   - **Behavior of \( g(x) \) as \( x \) approaches 2:**
     - From the graph of \( g(x) \), as \( x \) approaches 2 from both sides, the value of \( g(x) \) approaches 2.

   - **Limit Calculation:**
     \[
     \lim\limits_{x \to 2} \frac{f(x)}{g(x)} = \frac{1}{2}
     \]

2. **Limit 2:** \(\lim\limits_{x \to 2} \left[f(x) + g(x)\right]\)

   - **Behavior of \( f(x) \) as \( x \) approaches 2:**
     - As previously noted, \( f(x) \) approaches 1 as \( x \) approaches 2.

   - **Behavior of \( g(x) \) as \( x \) approaches 2:**
     - As previously noted, \( g(x) \) approaches 2 as \( x \) approaches 2.

   - **Limit Calculation:**
     \[
     \lim\limits_{x \to 2} \left[f(x) + g(x)\right] = 1 + 2 = 3
     \]

3. **Limit 3:** \(\lim\limits_{x \to 2} \frac{1}{g(x)}\)

   - **Behavior of \( g(x) \) as \( x \) approaches 2:**
     - \( g(x) \) approaches 2 as \( x \) approaches 2.

   - **Limit Calculation:**
     \[
     \lim\limits_{x \to 2} \frac{1}{g(x)} = \frac{1}{2}
     \]

4. **Limit 4:** \(\lim\limits_{x \to 2} f(x)g(x)\)

   - **Behavior of \( f(x) \) as \( x \) approaches 2:**
     - \( f(x) \) approaches 1 as \( x \) approaches 2.

   - **Behavior of \( g(x) \) as \( x \) approaches 2:**
     - \( g(x) \) approaches 2 as \( x \) approaches 2.

   - **Limit Calculation:**
     \[
     \lim\limits_{x \to 2} f(x)g(x) = 1 \times 2 = 2
     \]

**Summary of Limits:**
1. \(\lim\limits_{x \to 2} \frac{f(x)}{g(x)} = \frac{1}{2}\)
2. \(\lim\limits_{x \to 2} \left[f(x) + g(x)\right] = 3\)
3. \(\lim\limits_{x \to 2} \frac{1}{g(x)} = \frac{1}{2}\)
4. \(\lim\limits_{x \to 2} f(x)g(x) = 2\)

Would you like more details on any of these calculations or have any other questions?

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Here are 5 related questions that could expand your understanding:

1. What happens to the limits if \( g(x) \) approaches 0 as \( x \) approaches 2?
2. How would the limits change if \( f(x) \) approached 0 instead of 1 as \( x \) approaches 2?
3. What is the significance of \( g(x) \) having a vertical asymptote near \( x = 2 \)?
4. Can you calculate the limit of \( \frac{f(x) - g(x)}{x - 2} \) as \( x \) approaches 2?
5. How would you approach these limits if \( f(x) \) or \( g(x) \) were piecewise functions?

**Tip:** When analyzing limits, always ensure you understand the behavior of both the numerator and denominator as the variable approaches the point of interest, especially when dealing with fractions.

The graphs of f(x) and g(x) are given above. Use them to evaluate each quantity below. Write DNE if the limit or value does not exist (or if it’s infinity). 1. lim(x→2) f(x)/g(x) 2. lim(x→2) [f(x) + g(x)] 3. lim(x→2) 1/g(x) 4. lim(x→2) f(x)g(x)

Learn how to evaluate limits involving f(x) and g(x) as x approaches 2 using graphical analysis. This problem focuses on applying key calculus concepts, including limit laws and the behavior of functions as they approach a specific point. Suitable for high school students in Grades 11-12.

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