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

검증된 단계별 안내

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.

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.

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.

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