Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x² − 2x, x ≤ 1
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.53
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to 1^+} x^{\frac{1}{1 - x}}\).
Recognize that as \(x \to 1^+\), the base \(x\) approaches 1 and the exponent \(\frac{1}{1 - x}\) tends to \(+\infty\), creating an indeterminate form of type \(1^{\infty}\).
Rewrite the expression using the exponential and natural logarithm to handle the indeterminate form: \(x^{\frac{1}{1 - x}} = e^{\frac{1}{1 - x} \cdot \ln(x)}\).
Focus on finding the limit of the exponent: \(\lim_{x \to 1^+} \frac{\ln(x)}{1 - x}\). This is a \(\frac{0}{0}\) indeterminate form, so apply L'Hôpital's Rule by differentiating numerator and denominator separately.
After applying L'Hôpital's Rule, evaluate the resulting limit and then substitute back into the exponential expression 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 and Continuity
Limits describe the behavior of a function as the input approaches a particular value. Understanding how to evaluate limits, especially when the function approaches a point of discontinuity or an indeterminate form, is essential for analyzing the behavior near that point.
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Indeterminate Forms
Indeterminate forms like 0^0, ∞^0, or 1^∞ arise in limit problems where direct substitution does not yield a clear answer. Recognizing these forms helps in applying appropriate techniques such as logarithmic transformation or L'Hôpital's Rule to resolve the limit.
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Logarithmic Transformation for Limits
When limits involve expressions raised to variable powers, taking the natural logarithm can simplify the problem by converting the power into a product. This allows the use of limit laws and L'Hôpital's Rule on the exponent, after which exponentiation recovers the original limit.
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Related Practice
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45. y=cos(x-arccos(x))
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In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
23. y = ln(x)/(1+ln(x))
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Textbook Question
Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.
f(x) = 27x³
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Evaluate the integrals in Exercises 31–78.
73. ∫dx/√(-2x-x²)
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