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.

Verified step by step guidance

1. 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 \( \lim_{x \to a^-} f(x) \).
2. 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.
3. If \( \lim_{x \to a^+} f(x) = L \) and \( \lim_{x \to a^-} f(x) = L \), then \( \lim_{x \to a} f(x) = L \).
4. If \( \lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x) \), then \( \lim_{x \to a} f(x) \) does not exist.
5. 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.

Key Concepts

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.