Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>


h. ${\lim }_{x\to 3}f\left(x\right)$

Verified step by step guidance

1. Identify the point of interest, which is x = 3, on the graph of the function f(x).
2. Examine the behavior of the function as x approaches 3 from the left (x -> 3^-).
3. Examine the behavior of the function as x approaches 3 from the right (x -> 3^+).
4. Compare the left-hand limit and the right-hand limit to determine if they are equal.
5. If the left-hand limit and right-hand limit are equal, the limit exists and is equal to that common value; otherwise, state that the limit does not exist and explain why.

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. For example, the limit of f(x) as x approaches 3 indicates what value f(x) is approaching when x gets very close to 3, regardless of the actual value of f(3).

Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. This concept is crucial for determining whether a limit exists. If there is a jump, hole, or vertical asymptote in the graph at x = 3, the limit may not exist, indicating a discontinuity.

Existence of Limits

The existence of a limit at a point requires that the left-hand limit and right-hand limit both approach the same value. If these two limits differ or if one of them does not exist, then the overall limit does not exist. Understanding this concept is vital for analyzing the graph of f(x) near x = 3 to determine the limit's existence.