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>


d. ${\lim }_{x\to 1}f\left(x\right)$

Verified step by step guidance

1. Identify the behavior of the function \( f(x) \) as \( x \) approaches 1 from both the left and the right.
2. Examine the graph to determine the value that \( f(x) \) approaches as \( x \to 1^- \) (from the left).
3. Examine the graph to determine the value that \( f(x) \) approaches as \( x \to 1^+ \) (from the right).
4. Compare the left-hand limit and the right-hand limit. If they are equal, the limit exists and is equal to that common value.
5. If the left-hand limit and the right-hand limit are not equal, state that the limit does not exist and explain that the function approaches different values from the left and right.

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 near points of interest. For example, the limit of f(x) as x approaches 1 examines the values f(x) takes as x gets closer to 1, which can indicate whether f is defined or behaves predictably at that point.

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 asymptote at the point, the limit may not exist, indicating a discontinuity in the function.

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, the limit does not exist. Understanding this concept is vital for analyzing the graph of f(x) at x = 1, as it helps identify whether the function approaches a specific value from both sides or if there are discrepancies that prevent a limit from being defined.