Use the graph of g(x) in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>
${\lim }_{x\to 4}g\left(x\right)$

Verified step by step guidance

1. Identify the behavior of the function g(x) as x approaches 4 from the left (x -> 4^-).
2. Identify the behavior of the function g(x) as x approaches 4 from the right (x -> 4^+).
3. Compare the left-hand limit and the right-hand limit of g(x) as x approaches 4.
4. If both the left-hand limit and the right-hand limit are equal, then the limit $\underset{x\to 4}{lim}g(x)$ exists and is equal to that common value.
5. If the left-hand limit and the right-hand limit are not equal, then the limit $\underset{x\to 4}{lim}g(x)$ does not exist.

Key Concepts

Limits

A limit describes the behavior of a function as its input approaches a certain value. It is essential for understanding continuity and the behavior of functions at specific points. In this context, we are interested in the limit of g(x) as x approaches 4, which can indicate whether g(x) approaches a specific value or diverges.

Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. For the limit of g(x) as x approaches 4 to exist, g(4) must also be defined and equal to the limit. Discontinuities can arise from jumps, holes, or vertical asymptotes in the graph.

Existence of Limits

The existence of a limit requires that the left-hand limit and right-hand limit at a point are equal. If they differ or if one of them does not exist, then the overall limit does not exist. In analyzing g(x) at x = 4, one must check the behavior of the function from both sides of 4 to determine if the limit exists.