The function 
\[g\] is defined for all real numbers.
Function g is graphed. The x-axis goes from negative 8 to 8. The graph consists of a curve and a closed circle. The curve starts at (negative 8, 2), moves upward to (negative 4, 4), and moves downward through an open circle at x = 2, midway between y = 1 and y = 2. The curve ends in quadrant 1. The closed circle is plotted at x = 2, between y = negative 2 and y = negative 3, closer to y = negative 2.
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[y\]
\[x\]
\[\blueD g\]
What is a reasonable estimate for  
 
\[\displaystyle\lim_{x\to 2}g(x)\] ?
Choose 1 answer:
Choose 1 answer:
(Choice A)   
\[-2.4\]
A
\[-2.4\]
(Choice B)   
\[-2\]
B
\[-2\]
(Choice C)   
\[1.5\]
C
\[1.5\]
(Choice D)   
\[2\]
D
\[2\]
(Choice E)   The limit doesn't exist.
E
The limit doesn't exist.

The question is asking for the value of the limit \(\lim_{x \to 2} g(x)\), which refers to the value that the function \(g(x)\) approaches as \(x\) gets closer to 2 from both sides (left and right).

### Key points from the description of the graph:
- There's an open circle at \(x = 2\) between \(y = 1\) and \(y = 2\).
- The curve approaches this open circle as \(x\) approaches 2 from both the left and right sides.
- A closed circle is plotted at \(x = 2\) near \(y = -2.4\), but this is the actual value of the function at \(x = 2\), not the limit.

The function’s value at \(x = 2\) (which is \(-2.4\)) doesn't affect the limit, because the limit is determined by the values that the function approaches from both sides of \(x = 2\).

Since the open circle (which represents the value the function approaches) is midway between \(y = 1\) and \(y = 2\), a reasonable estimate for \(\lim_{x \to 2} g(x)\) is \(1.5\).

### Conclusion:
The correct answer is:
\[
\boxed{1.5}
\]

Would you like more details on how limits work, or have any other questions?

### Related Questions:
1. What is the difference between a limit and the function value?
2. How does an open circle on a graph affect the limit?
3. Why is the function value at \(x = 2\) irrelevant for the limit in this case?
4. How can limits be estimated from a graph?
5. What are left-hand and right-hand limits?

### Tip:
Always check whether the limit exists from both directions. If the values from the left and right are different, the limit does not exist.

The function g is defined for all real numbers. Function g is graphed. The x-axis goes from negative 8 to 8. The graph consists of a curve and a closed circle. The curve starts at (negative 8, 2), moves upward to (negative 4, 4), and moves downward through an open circle at x = 2, midway between y = 1 and y = 2. The curve ends in quadrant 1. The closed circle is plotted at x = 2, between y = negative 2 and y = negative 3, closer to y = negative 2. What is a reasonable estimate for \(\lim_{x\to 2}g(x)\)?

Understand how to estimate the limit \(\lim_{x\to 2} g(x)\) using graph analysis. This problem explores key calculus concepts such as left-hand and right-hand limits and how they influence the existence of a limit. Suitable for high school students in Grades 10-12.

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