Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problema 4.R.67

Capitolo 4, Problema 4.R.67

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_y→0⁺ (ln¹⁰ y) / √y

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First, identify the form of the limit as y approaches 0 from the positive side. The expression (ln¹⁰ y) / √y becomes (-∞) / 0⁺, which is an indeterminate form.

Since the limit is in an indeterminate form, apply l'Hôpital's Rule. This rule states that if the limit of f(y)/g(y) as y approaches a point is indeterminate, then the limit is the same as the limit of f'(y)/g'(y) as y approaches that point, provided the derivatives exist.

Differentiate the numerator, ln¹⁰ y, with respect to y. Use the chain rule: the derivative of ln¹⁰ y is 10 * ln⁹ y * (1/y).

Differentiate the denominator, √y, with respect to y. The derivative of √y is (1/2) * y^(-1/2).

Apply l'Hôpital's Rule by taking the limit of the new expression: lim_y→0⁺ [10 * ln⁹ y * (1/y)] / [(1/2) * y^(-1/2)]. Simplify the expression and evaluate the limit as y approaches 0 from the positive side.

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Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

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One-Sided Limits

L'Hôpital's Rule

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

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Power Rules

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in limits and integrals, due to its unique properties, such as the fact that the derivative of ln(x) is 1/x. Understanding the behavior of ln(x) as x approaches 0 is essential for evaluating limits involving logarithmic functions.

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Derivative of the Natural Logarithmic Function

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60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.