To determine $\lim_{{x \to 0.5}} g(x)$, we need to analyze the behavior of the function $g(x)$ as $x$ approaches 0.5 from both the left and the right, based on the provided graphical information.

Given:

• There is a filled circle at $x = 0.5$, $y = 1$.
• There is an open circle at $x = 0.5$, $y = 1.5$.

Interpretation:

• The filled circle at $x = 0.5$, $y = 1$ means that the value of the function $g(x)$ at $x = 0.5$ is $g(0.5) = 1$.
• The open circle at $x = 0.5$, $y = 1.5$ suggests that the function approaches $y = 1.5$ as $x$ approaches 0.5, but does not actually take this value at $x = 0.5$.

Limit evaluation:

To determine the limit $\lim_{{x \to 0.5}} g(x)$, we focus on the values of the function as $x$ gets arbitrarily close to 0.5 from either direction:

• As $x \to 0.5^-$ (from the left) and $x \to 0.5^+$ (from the right), the function approaches the height $y = 1.5$, as indicated by the open circle.

Therefore, the limit is:

$\lim_{{x \to 0.5}} g(x) = 1.5$

Summary:

• The value of the function at $x = 0.5$ is $g(0.5) = 1$.
• The limit as $x$ approaches 0.5 is $\lim_{{x \to 0.5}} g(x) = 1.5$.

Would you like more details or clarifications? Here are some questions to deepen your understanding:

1. What does it mean when a circle is filled or open on a graph?
2. How do one-sided limits differ from two-sided limits?
3. Can a function have a limit at a point where it is not defined?
4. How would the limit change if the open circle was at a different $y$-value?
5. How can a function be continuous at a point? Is $g(x)$ continuous at $x = 0.5$?

Tip: Open circles indicate points where the function does not take the given value, but the function can still approach that value as a limit.