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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.7.31
Chapter 4, Problem 4.7.31

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_u→ π/4 (tan u - cot u) / (u - π/4)

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Identify the form of the limit as u approaches π/4. Substitute u = π/4 into the expression (tan u - cot u) / (u - π/4) to check if it results in an indeterminate form like 0/0.
Since substituting u = π/4 results in an indeterminate form 0/0, l'Hôpital's Rule is applicable. According to l'Hôpital's Rule, if the limit of f(u)/g(u) as u approaches a value results in 0/0 or ∞/∞, then the limit can be evaluated as the limit of f'(u)/g'(u).
Differentiate the numerator and the denominator separately. The numerator is tan(u) - cot(u). Differentiate tan(u) to get sec^2(u) and differentiate cot(u) to get -csc^2(u).
Differentiate the denominator, which is simply u - π/4. The derivative of u is 1, and the derivative of a constant π/4 is 0.
Apply l'Hôpital's Rule by taking the limit of the new expression: lim_{u→π/4} (sec^2(u) + csc^2(u)) / 1. Evaluate this limit by substituting u = π/4 into the expression.

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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 continuity and the value that a function approaches, even if it is not explicitly defined at that point. Evaluating limits is crucial for determining the behavior of functions near points of interest, especially in cases of indeterminate forms.
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L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in 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 separately. This rule simplifies the process of finding limits and is particularly useful in calculus when direct substitution fails.
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Trigonometric Functions

Trigonometric functions, such as tangent (tan) and cotangent (cot), are essential in calculus for analyzing periodic behavior and angles. The tangent function is defined as the ratio of the sine and cosine functions, while cotangent is the reciprocal of tangent. Understanding their properties and how they behave near specific angles, like π/4, is crucial for evaluating limits involving these functions.
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Introduction to Trigonometric Functions
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