112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)

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Recall the definition of the little-o notation: \(f(x) = o(g(x))\) as \(x \to a\) means that \(\lim_{x \to a} \frac{f(x)}{g(x)} = 0\).

Identify the functions in the expression: here, \(f(x) = \ln x\) and \(g(x) = x + 1\).

Determine the point at which the limit is taken. Since it is not explicitly stated, consider the common limit point for \(\ln x\), which is \(x \to 0^+\) or \(x \to 1\); check both if necessary.

Compute the limit \(\lim_{x \to a} \frac{\ln x}{x + 1}\) for the chosen \(a\) to verify if it equals zero.

Based on the limit result, conclude whether \(\ln x = o(x + 1)\) is true or false, providing reasoning from the limit behavior.

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주요 개념

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

Big-O Notation

Big-O notation describes the upper bound of a function's growth rate near a point, often used to compare how functions behave as the input approaches a limit. It provides a way to express that one function grows no faster than another up to constant factors.

Behavior of the Natural Logarithm Near x = -1

The function ln(x) is only defined for x > 0, so evaluating ln(x) near x = -1 is not possible in the real number domain. Understanding the domain restrictions is crucial when analyzing limits or asymptotic behavior involving logarithms.

Limit and Asymptotic Equivalence

To say f(x) = O(g(x)) as x approaches a point means that |f(x)| ≤ C|g(x)| near that point for some constant C. This requires analyzing the limit behavior of the ratio f(x)/g(x) to determine if it remains bounded.

112. True, or false? Give reasons for your answers.c. ln x = o(x+1)