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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 4.7.54
Chapter 4, Problem 4.7.54

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


lim_x→1⁻ (1-x) tan πx/2

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First, identify the form of the limit as x approaches 1 from the left. Substitute x = 1 into the expression (1-x) tan(πx/2) to check if it results in an indeterminate form like 0/0 or ∞/∞.
Notice that as x approaches 1 from the left, (1-x) approaches 0 and tan(πx/2) approaches tan(π/2), which is undefined. However, from the left, tan(πx/2) approaches negative infinity, creating an indeterminate form of 0 * (-∞).
To apply l'Hôpital's Rule, rewrite the expression as a fraction. Consider the limit of (1-x) / (cot(πx/2)), since cotangent is the reciprocal of tangent.
Now, check if the rewritten expression (1-x) / (cot(πx/2)) results in a 0/0 form as x approaches 1 from the left. If it does, l'Hôpital's Rule can be applied.
Apply l'Hôpital's Rule by differentiating the numerator and the denominator separately. Differentiate (1-x) to get -1, and differentiate cot(πx/2) using the chain rule to get -π/2 * csc²(πx/2). Then, evaluate the limit of the new expression as x approaches 1 from the left.

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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.
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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 ∞/∞. The rule 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 process can be repeated if the result remains indeterminate.
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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. In the context of limits, these functions can exhibit specific behaviors as their arguments approach certain values, which can lead to indeterminate forms. Understanding the properties and limits of these functions is essential for evaluating limits involving trigonometric expressions.
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Introduction to Trigonometric Functions
Related Practice
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