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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.R.75
Chapter 4, Problem 4.R.75

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 
lim_x→π / 2- (sin x) ^tan x

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Step 1: Identify the form of the limit as x approaches π/2 from the left. The expression (sin x)^tan x can be rewritten using the exponential function: e^(tan x * ln(sin x)).
Step 2: Analyze the behavior of sin x and tan x as x approaches π/2 from the left. Note that sin x approaches 1 and tan x approaches infinity.
Step 3: Recognize that the limit involves an indeterminate form of type 1^∞. To resolve this, take the natural logarithm of the expression, which transforms the problem into finding the limit of tan x * ln(sin x) as x approaches π/2 from the left.
Step 4: Apply l'Hôpital's Rule to the limit of tan x * ln(sin x). This requires differentiating the numerator and the denominator separately. The derivative of tan x is sec^2 x, and the derivative of ln(sin x) is cot x.
Step 5: Evaluate the new limit using l'Hôpital's Rule. If necessary, apply l'Hôpital's Rule multiple times until the limit can be determined. Once the limit of the logarithmic expression is found, exponentiate the result to find the original limit.

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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 in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches π/2 involves analyzing the behavior of the function near that point, which may lead to indeterminate forms.
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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 these forms occur, the limit of the ratio of two functions 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 when direct substitution is not possible.
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Trigonometric Functions

Trigonometric functions, such as sine and tangent, are periodic functions that relate angles to ratios of sides in right triangles. Understanding their behavior, especially near critical points like π/2, is crucial for limit evaluation. In this problem, the sine function approaches 1 as x approaches π/2, while the behavior of tan x must also be considered to determine the overall limit.
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Related Practice
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