determine lim (x->0.5) g(x) for g as in the figure. Above 0.5 at y=1 is a filled circle and at y=1.5 there is an open circle.

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.

Learn how to determine the limit of a function as x approaches 0.5, using graphical interpretation with filled and open circles. The problem explores the concept of limits, continuity, and how open circles indicate the behavior of a function near a point.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation