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

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.4.13

Chapter 2, Problem 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).

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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