Theory and Examples

a. If limx→0 f(x) / x² = 1, find limx→0 f(x).

검증된 단계별 안내

First, understand the given limit: \( \lim_{x \to 0} \frac{f(x)}{x^2} = 1 \). This means that as \( x \) approaches 0, the function \( \frac{f(x)}{x^2} \) approaches 1.

To find \( \lim_{x \to 0} f(x) \), consider the behavior of \( f(x) \) as \( x \) approaches 0. Since \( \frac{f(x)}{x^2} \to 1 \), it implies that \( f(x) \) behaves like \( x^2 \) near 0.

Multiply both sides of the equation \( \frac{f(x)}{x^2} = 1 \) by \( x^2 \) to isolate \( f(x) \): \( f(x) = x^2 \cdot 1 = x^2 \).

Now, substitute \( f(x) = x^2 \) into the limit expression: \( \lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 \).

Evaluate the limit \( \lim_{x \to 0} x^2 \). As \( x \to 0 \), \( x^2 \to 0 \). Therefore, \( \lim_{x \to 0} f(x) = 0 \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of f(x) as x approaches 0. Understanding limits is crucial for analyzing the continuity and behavior of functions at specific points.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can simplify the process of finding limits in complex scenarios.

Continuous Functions

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. In this problem, if we find that limx→0 f(x) exists and equals a specific value, it indicates that f(x) is continuous at x = 0, which is essential for understanding the behavior of the function around that point.

Theory and Examplesa. If limx→0 f(x) / x² = 1, find limx→0 f(x).