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>
${\lim }_{x\to 1^{+}}f\left(x\right)$

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. For example, the limit of f(x) as x approaches 1 from the right (denoted as lim x→1⁺ f(x)) examines the values f(x) takes as x gets closer to 1 from values greater than 1.

One-Sided Limits

One-sided limits are specific types of limits that consider the behavior of a function as the input approaches a particular value from one side only. The right-hand limit (lim x→1⁺ f(x)) looks at values approaching from the right, while the left-hand limit (lim x→1⁻ f(x)) considers values approaching from the left. Understanding one-sided limits is crucial for determining the overall limit at a point, especially when the function exhibits different behaviors from each side.

Existence of Limits

A limit exists if both the left-hand and right-hand limits at a point are equal. If they differ, the limit does not exist. Additionally, if the function approaches infinity or oscillates without settling at a value, the limit is also considered non-existent. Analyzing the existence of limits is essential for understanding continuity and the overall behavior of functions at specific points.