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?
e. x

검증된 단계별 안내

Recall that the growth rate of functions as \(x \to \infty\) can be compared using limits of their ratios. Specifically, for two functions \(f(x)\) and \(g(x)\), if \(\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty\), then \(f(x)\) grows faster than \(g(x)\). If the limit is a finite nonzero constant, they grow at the same rate. If the limit is 0, \(f(x)\) grows slower than \(g(x)\).

Identify the functions to compare: here, \(f(x) = x\) and \(g(x) = \ln(x)\).

Set up the limit to compare their growth rates: \(\lim_{x \to \infty} \frac{x}{\ln(x)}\).

Analyze the limit: since \(x\) grows without bound much faster than \(\ln(x)\), this limit tends to \(\infty\), indicating that \(x\) grows faster than \(\ln(x)\) as \(x \to \infty\).

Conclude that the function \(x\) grows faster than \(\ln(x)\) as \(x\) approaches infinity.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 vs. Polynomial Growth

Logarithmic functions like ln(x) grow very slowly compared to polynomial functions such as x. As x approaches infinity, polynomial functions increase much faster than logarithmic ones, making them dominant in growth comparisons.

Asymptotic Behavior and Limits

Asymptotic behavior studies the trend of functions as x approaches infinity. Using limits, we can compare the ratio of two functions to determine if one grows faster, slower, or at the same rate as the other.

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?e. x