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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.P.112c
Chapter 7, Problem 7.P.112c

112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)

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1
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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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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Derivative of the Natural Logarithmic Function

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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Asymptotes of Hyperbolas
Related Practice
Textbook Question

112. True, or false? Give reasons for your answers.

a. 1/x⁴ = O(1/x² + 1/x⁴)

Textbook Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

5. y = ln(sin²θ)

Textbook Question

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

99. lim(x→0) (2^sin(x) - 1)/(e^x - 1)

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

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2

Textbook Question

111. True, or false? Give reasons for your answers.

c. x = o(x + ln(x))