Suppose f(x)→100 and g(x)→0, with g(x)\u003c0 as x→2. Determine lim x→2 f(x) / g(x).

Ch. 2 - Limits

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Briggs 3rd Edition

Ch. 2 - Limits

문제 2.4.13

2장, 문제 2.4.13

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

검증된 단계별 안내

Identify the given limits: \( \lim_{x \to 2} f(x) = 100 \) and \( \lim_{x \to 2} g(x) = 0 \).

Recognize that \( g(x) < 0 \) as \( x \to 2 \), indicating that \( g(x) \) approaches zero from the negative side.

Apply the limit of a quotient rule: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) if the limits exist and \( \lim_{x \to a} g(x) \neq 0 \).

Since \( \lim_{x \to 2} g(x) = 0 \), the quotient \( \frac{f(x)}{g(x)} \) approaches an indeterminate form of \( \frac{100}{0} \).

Conclude that \( \lim_{x \to 2} \frac{f(x)}{g(x)} \) is undefined and approaches \( -\infty \) because \( g(x) \) approaches zero from the negative side.

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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. In this case, we are interested in the limits of f(x) and g(x) as x approaches 2. Understanding limits is crucial for evaluating expressions that may not be directly computable at a specific point.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to ambiguous results, such as 100/0. In this scenario, since f(x) approaches 100 and g(x) approaches 0 from the negative side, we need to analyze the limit further to determine the behavior of the quotient.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms by differentiating the numerator and denominator. If the limit results in a form like 100/0, applying this rule can help find the limit of the quotient by examining the derivatives of f(x) and g(x) as x approaches 2.