Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
5. y = ln(sin²θ)
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.P.112c
112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)
Verified step by step guidance1
Recall the definition of the little-o notation: \(f(x) = o(g(x))\) as \(x \to a\) means that \(\lim_{x \to a} \frac{f(x)}{g(x)} = 0\).
Identify the functions in the expression: here, \(f(x) = \ln x\) and \(g(x) = x + 1\).
Determine the point at which the limit is taken. Since it is not explicitly stated, consider the common limit point for \(\ln x\), which is \(x \to 0^+\) or \(x \to 1\); check both if necessary.
Compute the limit \(\lim_{x \to a} \frac{\ln x}{x + 1}\) for the chosen \(a\) to verify if it equals zero.
Based on the limit result, conclude whether \(\ln x = o(x + 1)\) is true or false, providing reasoning from the limit behavior.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Big-O Notation
Big-O notation describes the upper bound of a function's growth rate near a point, often used to compare how functions behave as the input approaches a limit. It provides a way to express that one function grows no faster than another up to constant factors.
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04:22
Sigma Notation
Behavior of the Natural Logarithm Near x = -1
The function ln(x) is only defined for x > 0, so evaluating ln(x) near x = -1 is not possible in the real number domain. Understanding the domain restrictions is crucial when analyzing limits or asymptotic behavior involving logarithms.
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Limit and Asymptotic Equivalence
To say f(x) = O(g(x)) as x approaches a point means that |f(x)| ≤ C|g(x)| near that point for some constant C. This requires analyzing the limit behavior of the ratio f(x)/g(x) to determine if it remains bounded.
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Related Practice
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Textbook Question
Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?
Textbook Question
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)
Textbook Question
7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?