Use the graphs of f and g in the accompanying figure tofind the limits that exist. If the limit does not exist, explainwhy.(a) limx → 2 [f(x) + g(x)]

Since there is no accompanying figure for the graphs of $f$ and $g$, I'll explain how to approach the problem step-by-step to find the limit $\lim_{x \to 2} [f(x) + g(x)]$.

1. ### Break Down the Problem

1. Determine the value of $f(2)$ if it exists.
2. Determine the value of $g(2)$ if it exists.
3. Calculate $f(2) + g(2)$ to find the limit.

2. ### Relevant Concepts

1. The limit $\lim_{x \to a} [f(x) + g(x)]$ exists if both $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist.

3. ### Analysis and Detail

1. Check if $f(x)$ approaches a specific value as $x$ approaches 2.
2. Check if $g(x)$ approaches a specific value as $x$ approaches 2.
3. If both limits $\lim_{x \to 2} f(x)$ and $\lim_{x \to 2} g(x)$ exist, then sum the results.

4. ### Verify and Summarize

1. If either limit does not exist, then $\lim_{x \to 2} [f(x) + g(x)]$ does not exist.
2. If both limits exist, then combine the values.

Final Answer

Due to the absence of the graph, I cannot provide definitive numeric results. Depending on the behavior of $f$ and $g$ at $x = 2$:

• If $f(2)$ and $g(2)$ exist, then $\lim_{x \to 2} [f(x) + g(x)] = f(2) + g(2)$.
• If either $f(2)$ or $g(2)$ does not exist (due to discontinuities or undefined values), then $\lim_{x \to 2} [f(x) + g(x)]$ does not exist.

Please check the graph of $f$ and $g$ at $x = 2$ for specific limits.