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?
g. ln(ln x)

검증된 단계별 안내

Recall that when comparing growth rates of functions as \(x \to \infty\), we often use limits of their ratios to determine which grows faster, slower, or at the same rate.

Identify the two functions to compare: \(f(x) = \ln(x)\) and \(g(x) = \ln(\ln x)\), where \(x\) is large enough so that \(\ln x > 0\).

Consider the limit \(\lim_{x \to \infty} \frac{g(x)}{f(x)} = \lim_{x \to \infty} \frac{\ln(\ln x)}{\ln x}\). This limit will help us understand their relative growth rates.

Analyze the behavior of the limit: since \(\ln x\) grows without bound but more slowly than any power of \(x\), and \(\ln(\ln x)\) grows even more slowly, the numerator grows much slower than the denominator.

Conclude that because the limit tends to zero, \(g(x) = \ln(\ln x)\) grows slower than \(f(x) = \ln x\) as \(x \to \infty\).

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주요 개념

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Growth Rates of Functions

Growth rates describe how functions behave as their input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace as others, especially as x approaches infinity.

Logarithmic Functions and Their Properties

Logarithmic functions like ln(x) grow slowly compared to polynomial or exponential functions. Understanding properties of ln(x) and nested logarithms such as ln(ln x) is essential to compare their growth rates accurately.

Asymptotic Comparison Using Limits

To compare growth rates, limits of ratios of functions as x approaches infinity are used. If the limit of f(x)/g(x) is zero, f grows slower; if infinite, f grows faster; if finite and nonzero, they grow at the same rate.

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?g. ln(ln x)