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>
$lim over x right arrow 4 of g of x$

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Identify the behavior of the function g(x) as x approaches 4 from the left (x -> 4^-).

Identify the behavior of the function g(x) as x approaches 4 from the right (x -> 4^+).

Compare the left-hand limit and the right-hand limit of g(x) as x approaches 4.

If both the left-hand limit and the right-hand limit are equal, then the limit $ \lim_{x\to4}g(x) $ exists and is equal to that common value.

If the left-hand limit and the right-hand limit are not equal, then the limit $ \lim_{x\to4}g(x) $ does not exist.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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