Textbook Question
What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?
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 7.8.6c
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
c. 1/√x
Verified step by step guidance1
Recall that the natural logarithm function \(\ln(x)\) grows without bound as \(x \to \infty\), but it does so very slowly compared to many other functions.
Consider the function given: \(f(x) = \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}\). As \(x \to \infty\), this function approaches zero because the denominator grows without bound.
To compare growth rates, analyze the limit of the ratio \(\frac{f(x)}{\ln(x)} = \frac{1/\sqrt{x}}{\ln(x)} = \frac{1}{\sqrt{x} \cdot \ln(x)}\) as \(x \to \infty\).
Since \(\sqrt{x}\) and \(\ln(x)\) both grow to infinity, their product \(\sqrt{x} \cdot \ln(x)\) also grows to infinity, making the ratio approach zero.
Because the ratio \(\frac{f(x)}{\ln(x)} \to 0\), the function \(\frac{1}{\sqrt{x}}\) grows slower than \(\ln(x)\) as \(x \to \infty\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Functions as x Approaches Infinity
Understanding how functions behave as x approaches infinity is essential for comparing their growth rates. Functions can grow faster, slower, or at the same rate depending on how quickly their values increase. For example, polynomial functions grow faster than logarithmic functions, while functions approaching zero grow slower.
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Integrals of Natural Exponential Functions (e^x) Example 3
Behavior of the Natural Logarithm Function ln(x)
The natural logarithm function ln(x) increases without bound as x approaches infinity, but it does so very slowly compared to polynomial or exponential functions. Recognizing the slow growth of ln(x) helps in comparing it with other functions to determine relative growth rates.
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05:18Derivative of the Natural Logarithmic Function
Limits and Comparison of Functions Using Limits
To compare growth rates, we often use limits of ratios of functions as x approaches infinity. If the limit of f(x)/g(x) is zero, f grows slower than g; if it is infinity, f grows faster; if it is a finite nonzero constant, they grow at the same rate. This method provides a precise way to classify growth behavior.
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Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. c. arcsec(-2)
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
c. lim(x→∞) sinh x
1
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Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. c. arccos(√3/2)
Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
c. y = e^(-x) + Ce^(-(3/2)x)