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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 7.8.10.c
Chapter 7, Problem 7.8.10.c

10. True, or false? As x→∞,
c. 1/x - 1/x² = o(1/x)

Verified step by step guidance
1
Recall the definition of the little-o notation: for functions f(x) and g(x), we say f(x) = o(g(x)) as x → ∞ if \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0 \).
Identify the functions in the problem: \( f(x) = \frac{1}{x} - \frac{1}{x^2} \) and \( g(x) = \frac{1}{x} \).
Form the ratio \( \frac{f(x)}{g(x)} = \frac{\frac{1}{x} - \frac{1}{x^2}}{\frac{1}{x}} \).
Simplify the ratio: \( \frac{\frac{1}{x} - \frac{1}{x^2}}{\frac{1}{x}} = \frac{1}{x} \cdot \frac{x}{1} - \frac{1}{x^2} \cdot \frac{x}{1} = 1 - \frac{1}{x} \).
Evaluate the limit as \( x \to \infty \): \( \lim_{x \to \infty} \left(1 - \frac{1}{x}\right) = 1 \). Since this limit is not zero, conclude that \( \frac{1}{x} - \frac{1}{x^2} \neq o\left(\frac{1}{x}\right) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Little-o Notation

Little-o notation, written as f(x) = o(g(x)) as x→∞, means that f(x) becomes insignificant compared to g(x) in the limit. Formally, f(x)/g(x) → 0 as x→∞, indicating f(x) grows much slower than g(x). It is used to describe the relative growth rates of functions.
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Asymptotic Behavior of Rational Functions

As x approaches infinity, terms like 1/x and 1/x² approach zero, but at different rates. Specifically, 1/x² tends to zero faster than 1/x, which affects how their differences behave asymptotically. Understanding these rates is key to comparing functions using little-o notation.
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Limit of a Quotient for Comparing Growth Rates

To determine if f(x) = o(g(x)), evaluate the limit of f(x)/g(x) as x→∞. If the limit is zero, f(x) is little-o of g(x). This method helps verify statements about asymptotic dominance, such as whether 1/x - 1/x² = o(1/x).
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Related Practice
Textbook Question

1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

c. √x

Textbook Question

c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).

Textbook Question

In Exercises 41–44:


c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that

(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a


44. f(x) = 2x², x ≥ 0, a = 5

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

Textbook Question

23. Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years.

c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?

Textbook Question

88. Given that x>0, find the maximum value, if any, of

c. x^(1/x^n) (n a positive integer)