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?

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.