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)

검증된 단계별 안내

Recall that the growth rate of functions as \(x \to \infty\) can be compared by analyzing their dominant terms or by using limits of their ratios.

Rewrite the given function \(\ln(\sqrt{x})\) using logarithm properties: \(\ln(\sqrt{x}) = \ln(x^{1/2}) = \frac{1}{2} \ln(x)\).

Compare \(\ln(\sqrt{x})\) to \(\ln(x)\) by considering the ratio \(\frac{\ln(\sqrt{x})}{\ln(x)} = \frac{\frac{1}{2} \ln(x)}{\ln(x)} = \frac{1}{2}\).

Since the ratio approaches a positive constant (\(\frac{1}{2}\)), \(\ln(\sqrt{x})\) grows at the same rate as \(\ln(x)\), but scaled by a constant factor.

Therefore, \(\ln(\sqrt{x})\) neither grows faster nor slower than \(\ln(x)\) in terms of growth rate classification; they grow at the same rate.

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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 their dominant terms. This concept helps classify functions by their long-term behavior, such as logarithmic, polynomial, or exponential growth.

Properties of Logarithmic Functions

Logarithmic functions like ln(x) grow slowly compared to polynomials and exponentials. Key properties include the logarithm of a root, ln(√x) = (1/2)ln(x), which shows that scaling inside the log translates to a constant multiple outside. This helps compare growth rates by simplifying expressions.

Asymptotic Equivalence and Big-O Notation

Asymptotic equivalence means two functions grow at the same rate if their ratio approaches a nonzero constant as x→∞. Big-O notation formalizes this by describing upper bounds on growth. Recognizing when functions differ by constant multiples is crucial for determining if they grow at the same rate.

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)