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

Problem 4.R.63

Chapter 4, Problem 4.R.63

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²

Verified step by step guidance

First, identify the form of the limit as Θ approaches 0. The expression (3 sin² 2Θ) / Θ² is in the indeterminate form 0/0, which suggests that l'Hôpital's Rule can be applied.

Apply l'Hôpital's Rule, which states that if the limit of f(Θ)/g(Θ) as Θ approaches a value results in an indeterminate form, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.

Differentiate the numerator: The derivative of 3 sin² 2Θ with respect to Θ involves using the chain rule. First, differentiate sin² 2Θ to get 2 sin 2Θ * cos 2Θ, and then multiply by the derivative of 2Θ, which is 2. Therefore, the derivative of the numerator is 12 sin 2Θ * cos 2Θ.

Differentiate the denominator: The derivative of Θ² with respect to Θ is straightforward, which is 2Θ.

Now, evaluate the limit of the new expression (12 sin 2Θ * cos 2Θ) / (2Θ) as Θ approaches 0. Simplify the expression and check if further application of l'Hôpital's Rule is needed or if the limit can be directly evaluated.

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Key Concepts

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

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.

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 functions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In calculus, they are often involved in limits and derivatives due to their unique properties, such as periodicity and boundedness. Understanding their behavior, especially near critical points like 0, is essential for evaluating limits involving trigonometric expressions.

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