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

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Understand the notation: The expression \(x = o(x + \ln(x))\) means that \(x\) is a little-o of \(x + \ln(x)\) as \(x\) approaches some limit (usually \(\infty\) or 0). This implies that \(\lim_{x \to a} \frac{x}{x + \ln(x)} = 0\), where \(a\) is the point of interest.

Determine the limit point: Since \(\ln(x)\) is defined for \(x > 0\), the natural domain to consider is \(x \to \infty\) or \(x \to 0^+\). We will analyze the limit \(\lim_{x \to \infty} \frac{x}{x + \ln(x)}\).

Set up the limit: Write the ratio explicitly as \(\lim_{x \to \infty} \frac{x}{x + \ln(x)}\) and consider the behavior of numerator and denominator as \(x\) grows large.

Simplify the expression inside the limit by dividing numerator and denominator by \(x\): \(\lim_{x \to \infty} \frac{1}{1 + \frac{\ln(x)}{x}}\).

Evaluate the limit: Since \(\frac{\ln(x)}{x} \to 0\) as \(x \to \infty\), the limit becomes \(\frac{1}{1 + 0} = 1\), which is not zero. Therefore, \(x\) is not \(o(x + \ln(x))\) as \(x \to \infty\).

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Little-o Notation (o-notation)

Little-o notation describes the limiting behavior of a function relative to another as the input approaches a point, usually zero or infinity. Specifically, f(x) = o(g(x)) means that f(x) grows much slower than g(x), or equivalently, the limit of f(x)/g(x) as x approaches the point is zero.

Behavior of the Natural Logarithm Near Zero

The natural logarithm function ln(x) approaches negative infinity as x approaches zero from the right. This unbounded behavior affects limits and comparisons involving ln(x) near zero, making expressions like x + ln(x) behave differently than simple polynomial terms.

Limit Comparison for Composite Functions

To determine if an expression like x = o(x + ln(x)) holds, one must analyze the limit of the ratio x / (x + ln(x)) as x approaches zero. Understanding how to compare growth rates and evaluate such limits is essential to verify or refute statements involving little-o notation.

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