Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.57
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
57. lim (x → 0⁺) x^(-1/ln x)
Verified step by step guidance1
Recognize that the limit involves an indeterminate form of the type \(x^{f(x)}\) as \(x \to 0^+\). To handle this, rewrite the expression using the exponential and logarithm functions: \(x^{-1/\ln x} = e^{\ln\left(x^{-1/\ln x}\right)}\).
Apply the logarithm power rule inside the exponent: \(\ln\left(x^{-1/\ln x}\right) = -\frac{1}{\ln x} \cdot \ln x\).
Simplify the expression inside the exponent: \(-\frac{1}{\ln x} \cdot \ln x = -1\).
Rewrite the original limit as \(\lim_{x \to 0^+} e^{-1}\), since the exponent simplifies to a constant.
Since \(e^{-1}\) is a constant, the limit is simply \(e^{-1}\). Thus, the limit exists and equals this constant value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits Involving Indeterminate Forms
Limits that result in expressions like 0^0, ∞^0, or 1^∞ are called indeterminate forms. These require special techniques such as rewriting the expression or applying logarithms to evaluate the limit accurately.
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Integrals Involving Natural Logs: Substitution
Logarithmic Transformation for Limits
Taking the natural logarithm of a function can simplify complex limit expressions, especially those involving exponents. By converting powers into products, it allows the use of limit laws and L'Hôpital's Rule more effectively.
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L'Hôpital's Rule
L'Hôpital's Rule helps evaluate limits that yield indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator separately. It is often used after logarithmic transformation to find limits of complicated expressions.
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5:50Power Rules
Related Practice
Textbook Question
41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.
a. How much longer would it take the soup to cool to 35°C?
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))
Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
35. y = ln((x²+1)^5/√(1-x))
1
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Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
117. y = (√t)ᵗ