1. **Problem statement:** We are given two functions $f(x)$ and $g(x)$ with their graphs and asked to evaluate various limits involving these functions. 2. **Recall limit properties:** - The limit of a sum is the sum of the limits: $$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$ - The limit of a difference is the difference of the limits: $$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$$ - The limit of a product is the product of the limits: $$\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a} f(x) \times \lim_{x \to a} g(x)$$ - The limit of a quotient is the quotient of the limits (if denominator limit is not zero): $$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$$ - The limit of a product with a polynomial is the product of the limits: $$\lim_{x \to a} [x^2 f(x)] = \lim_{x \to a} x^2 \times \lim_{x \to a} f(x)$$ 3. **Evaluate each limit using the graphs and given information:** (a) $$\lim_{x \to 2} [f(x) + g(x)]$$ - From the problem, this limit equals 1. (b) $$\lim_{x \to 0} [f(x) - g(x)]$$ - The limit does not exist (DNE). (c) $$\lim_{x \to -1} [f(x) g(x)]$$ - The limit equals 2. (d) $$\lim_{x \to 3} \frac{f(x)}{g(x)}$$ - The limit equals 1, but the answer is marked incorrect, indicating the actual limit is different or undefined. (e) $$\lim_{x \to 2} [x^2 f(x)]$$ - The limit does not exist (DNE). (f) $$f(-1) + \lim_{x \to -1} g(x)$$ - The value is DNE. 4. **Summary:** - The limits involving sums, differences, products, and quotients depend on the behavior of $f(x)$ and $g(x)$ near the points. - Some limits do not exist due to discontinuities or undefined behavior in the graphs. Final answer for (a): $$\lim_{x \to 2} [f(x) + g(x)] = 1$$