We are tasked with determining $\lim_{x \to 0.5} g(x)$ based on the given graph in Figure 11.

Analysis of the Graph:

1. The graph shows the function $y = g(x)$.
2. At $x = 0.5$, there is an open circle on the curve, which suggests that $g(0.5)$ is not defined at this point.
3. However, we can observe the behavior of the function as $x$ approaches $0.5$ from both sides (left and right).

Left-hand limit $\lim_{x \to 0.5^-} g(x)$:

• As $x$ approaches $0.5$ from the left, the function value approaches 1 (the y-coordinate of the curve).

Right-hand limit $\lim_{x \to 0.5^+} g(x)$:

• Similarly, as $x$ approaches $0.5$ from the right, the function also approaches 1.

Since both the left-hand and right-hand limits are equal, we conclude:

$\lim_{x \to 0.5} g(x) = 1$

Let me know if you would like more details or have any questions!

Related Questions:

1. How do we find limits graphically when there's an open circle at a point?
2. What is the difference between left-hand and right-hand limits?
3. Why is the function not defined at $x = 0.5$ despite the limit existing?
4. How do we handle limits where the function approaches infinity or negative infinity?
5. What is the formal definition of a limit?

Tip: When the function has an open circle at a point but the limits from both sides exist and are equal, the limit at that point exists, even if the function is not defined there.