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. $lim over x right arrow 1 of f of x$

검증된 단계별 안내

Identify the behavior of the function $ f(x) $ as $ x $ approaches 1 from both the left and the right.

Examine the graph to determine the value that $ f(x) $ approaches as $ x \to 1^- $ (from the left).

Examine the graph to determine the value that $ f(x) $ approaches as $ x \to 1^+ $ (from the right).

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

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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. limx→1f(x)\(\lim\)_{x\(\to\)1}f\(\left\)(x\(\right\))