Which of the following statements about the function y=f(x) graphed here are true, and which are false?

a. limx→2 f(x) does not exist.

검증된 단계별 안내

Step 1: Understand the concept of a limit. The limit of a function as x approaches a certain value is the value that the function approaches as x gets closer to that value.

Step 2: Analyze the graph of the function y = f(x) around x = 2. Look for the behavior of the function as x approaches 2 from both the left and the right.

Step 3: Check if the left-hand limit (as x approaches 2 from the left) and the right-hand limit (as x approaches 2 from the right) exist and are equal. If they are equal, the limit exists; if not, the limit does not exist.

Step 4: If the graph shows a jump, vertical asymptote, or any discontinuity at x = 2, then the limit as x approaches 2 does not exist.

Step 5: Conclude whether the statement 'lim x→2 f(x) does not exist' is true or false based on your analysis of the graph.

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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 determine the value that a function approaches, which may not necessarily be the function's value at that point. Understanding limits is crucial for analyzing 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. If a function has a discontinuity, it may have a limit that does not exist or differs from the function's value. Recognizing continuity is essential for evaluating limits and understanding the overall behavior of functions.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. They can exhibit different behaviors in different intervals, which may lead to discontinuities. Analyzing piecewise functions is important for determining limits and understanding how the function behaves at specific points, especially where the definition changes.

Which of the following statements about the function y=f(x) graphed here are true, and which are false?a. limx→2 f(x) does not exist.