Textbook Question
Verify the integration formulas in Exercises 111–114.
111. ∫ (arctan x) / x² dx = ln x - 1/2 ln(1 + x²) - arctan x / x + C
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 7.5.63
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
63. lim (x → ∞) ((x + 2)/(x - 1))^x
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to \infty} \left( \frac{x + 2}{x - 1} \right)^x\).
Rewrite the base inside the parentheses to express it in a form that approaches 1 as \(x\) approaches infinity. For example, write \(\frac{x + 2}{x - 1} = \frac{x - 1 + 3}{x - 1} = 1 + \frac{3}{x - 1}\).
Recognize that the limit has the indeterminate form \(1^\infty\), which suggests using the exponential and logarithm transformation: rewrite the limit as \(\lim_{x \to \infty} e^{x \ln \left(1 + \frac{3}{x - 1} \right)}\).
Focus on evaluating the exponent limit: \(\lim_{x \to \infty} x \ln \left(1 + \frac{3}{x - 1} \right)\). Use the fact that \(\ln(1 + y) \approx y\) for small \(y\) to simplify the expression inside the limit.
Calculate the simplified limit of the exponent, then substitute back into the exponential function to find the overall limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits at Infinity
Limits at infinity describe the behavior of a function as the input variable grows without bound. Understanding how expressions behave as x approaches infinity helps determine the end behavior of functions, which is essential for evaluating limits like ((x + 2)/(x - 1))^x as x → ∞.
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Cases Where Limits Do Not Exist
Indeterminate Forms
Indeterminate forms occur when a limit expression does not directly yield a clear value, such as 1^∞, 0/0, or ∞/∞. Recognizing these forms is crucial because they require special techniques, like logarithms or L'Hôpital's Rule, to resolve the limit properly.
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Exponential Limits and Logarithmic Transformation
When limits involve expressions raised to variable powers, rewriting the expression using logarithms can simplify evaluation. By setting y = f(x)^g(x), taking the natural log, and then applying limit laws, one can find the limit of y by exponentiating the limit of the logarithm.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 41–60.
57. ∫(from 1 to 2)cosh(ln t)/t dt
1
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Textbook Question
37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?
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.
124. x^(sin y) = ln y
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(cost+lnt)
Textbook Question
132. What is special about the functions
f(x) = arcsin((1/√(x²+1)) and g(x)=arctan(1/x)?
Explain.