Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>
lim x→3 f(x)

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Identify the point of interest, which is x = 3, on the graph of the function f(x).

Observe the behavior of the function as x approaches 3 from both the left (x → 3^-) and the right (x → 3^+).

Check if the values of f(x) from the left and right sides are approaching the same value as x approaches 3.

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. For example, evaluating the limit of f(x) as x approaches 3 involves determining the value that f(x) approaches as x gets closer to 3.

Continuity

Continuity refers to a property of a function where it is uninterrupted and has no breaks, jumps, or holes at a given point. A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. Understanding continuity is crucial for evaluating limits, as discontinuities can affect the limit's existence and value.

Graphical Analysis

Graphical analysis involves interpreting the visual representation of a function to understand its behavior, including limits and continuity. By examining the graph of f(x), one can identify trends, such as approaching values and discontinuities, which are essential for evaluating limits. This method provides intuitive insights that complement algebraic calculations.

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>lim x→3 f(x)