Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>
lim x→1 f(x)

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Identify the point of interest on the graph, which is x = 1.

Observe the behavior of the function f(x) as x approaches 1 from both the left (x → 1⁻) and the right (x → 1⁺).

Check if the values of f(x) from the left and right sides are approaching the same value as x approaches 1.

If the left-hand limit and the right-hand limit are equal, then the limit exists and is equal to that common value.

If the left-hand limit and the right-hand limit are not equal, then the limit does not exist.

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주요 개념

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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 specific points, even if they are not defined at those points. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Continuity ensures that there are no breaks, jumps, or holes in the graph of the function, which is essential for evaluating limits and understanding the overall behavior of the function.

Graphical Analysis

Graphical analysis involves interpreting the visual representation of a function to understand its properties, such as limits, continuity, and behavior at specific points. By examining the graph, one can often determine the limit of a function as it approaches a certain value, which is particularly useful when the function is not easily evaluated algebraically.

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>lim x→1 f(x)