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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.8.6c
Capitolo 7, Problema 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

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1
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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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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Derivative 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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Percorso guidato
07:45
Limit Comparison Test
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Domanda del libro di testo

86. This exercise explores the difference between

lim(x→∞)(1 + 1/x²)^x

and

lim(x→∞)(1 + 1/x)^x = e

c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

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Domanda del libro di testo

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

3. c. sin^(-1)(-√3/2)

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Domanda del libro di testo

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

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Domanda del libro di testo

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)

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Domanda del libro di testo

5. 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. ln(√x)

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Domanda del libro di testo

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)

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