Theory and Examples

Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

검증된 단계별 안내

To determine \( \lim_{x \to a} f(x) \), we need to consider both the right-hand limit \( \lim_{x \to a^+} f(x) \) and the left-hand limit \( \lim_{x \to a^-} f(x) \).

The limit \( \lim_{x \to a} f(x) \) exists if and only if both the right-hand limit and the left-hand limit exist and are equal.

If \( \lim_{x \to a^+} f(x) = L \) and \( \lim_{x \to a^-} f(x) = L \), then \( \lim_{x \to a} f(x) = L \).

If \( \lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x) \), then \( \lim_{x \to a} f(x) \) does not exist.

Therefore, knowing both \( \lim_{x \to a^+} f(x) \) and \( \lim_{x \to a^-} f(x) \) allows us to determine \( \lim_{x \to a} f(x) \) only if they are equal.

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주요 개념

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Limits from the Right and Left

In calculus, the limit of a function as it approaches a point from the right (denoted as limx→a+ f(x)) and from the left (denoted as limx→a− f(x)) are crucial for understanding the behavior of the function near that point. If both one-sided limits exist and are equal, it suggests that the overall limit at that point may also exist.

Existence of a Limit

A limit exists at a point if the values of the function approach a specific number as the input approaches that point from both sides. For limx→a f(x) to exist, it is necessary that both limx→a+ f(x) and limx→a− f(x) not only exist but also be equal. If they differ, the overall limit does not exist.

Continuity and Discontinuity

A function is continuous at a point if the limit as it approaches that point equals the function's value at that point. Discontinuities can arise when the one-sided limits are not equal or when either limit does not exist. Understanding these concepts helps in determining the nature of the function at the point in question.

Theory and ExamplesOnce you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.