To determine
$\lim_{x \to 2^-} [f(x) + g(x)]$ we analyze the left-hand limits of $f(x)$ and $g(x)$ at $x = 2$ based on the given graphs.

Step 1: Left-Hand Limit of $f(x)$

• The red graph represents $f(x)$.
• As $x$ approaches 2 from the left ($x \to 2^-$), we observe the corresponding $y$-value on the red curve.
• It appears that $\lim_{x \to 2^-} f(x) = 1$.

Step 2: Left-Hand Limit of $g(x)$

• The blue graph represents $g(x)$.
• As $x$ approaches 2 from the left ($x \to 2^-$), we observe the corresponding $y$-value on the blue curve.
• It appears that $\lim_{x \to 2^-} g(x) = -1$.

Step 3: Summing the Limits

$\lim_{x \to 2^-} [f(x) + g(x)] = \lim_{x \to 2^-} f(x) + \lim_{x \to 2^-} g(x) = 1 + (-1) = 0.$

Thus,
$\lim_{x \to 2^-} [f(x) + g(x)] = 0.$

Would you like further clarification or verification?

Related Questions:

1. What is $\lim_{x \to 2^+} [f(x) + g(x)]$?
2. Are $f(x)$ and $g(x)$ continuous at $x = 2$?
3. What are the one-sided limits of $f(x)$ and $g(x)$ at $x = 1$?
4. What is the behavior of $f(x) \cdot g(x)$ as $x \to 2^-$?
5. Can we conclude that $\lim_{x \to 2} [f(x) + g(x)]$ exists?

Tip:

Always check both one-sided limits when evaluating $\lim_{x \to a} f(x)$, as discrepancies may indicate discontinuities.