To evaluate

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

we analyze the left-hand limit ($x \to 2^-$) of both functions $f(x)$ (red) and $g(x)$ (blue).

Step 1: Identify $\lim_{x \to 2^-} f(x)$

From the red graph, as $x$ approaches $2$ from the left, the function value approaches approximately -1.

Step 2: Identify $\lim_{x \to 2^-} g(x)$

From the blue graph, as $x$ approaches $2$ from the left, the function value approaches approximately -1.

Step 3: Compute the Sum

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

Thus, the left-hand limit is:

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

Would you like further clarification or details on this?

Related Questions:

1. What is $\lim_{x \to 2^+} [f(x) + g(x)]$?
2. Are there any discontinuities in $f(x) + g(x)$ at $x = 2$?
3. How would you describe the behavior of $f(x)$ and $g(x)$ separately at $x=2$?
4. What does the sum of two discontinuous functions tell us about their combined behavior?
5. Can we conclude that $f(x) + g(x)$ is continuous at $x=2$?

Tip:

When computing limits graphically, always check for open and closed circles at the approach points to determine left-hand and right-hand behavior.