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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 7 - Transcendental Functions문제 7.5.66
7장, 문제 7.5.66

Indeterminate Powers and Products
Find the limits in Exercises 53–68.
66. lim (x → 0⁺) x (ln x)²

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Identify the form of the limit as \(x \to 0^+\) for the expression \(x (\ln x)^2\). Notice that as \(x\) approaches \(0\) from the right, \(x\) approaches \(0\) and \(\ln x\) approaches \(-\infty\), so \((\ln x)^2\) approaches \(+\infty\). This creates an indeterminate form of type \(0 \cdot \infty\).
Rewrite the expression to transform the product into a quotient, which is often easier to analyze with L'Hôpital's Rule. For example, write \(x (\ln x)^2\) as \(\frac{(\ln x)^2}{\frac{1}{x}}\).
Check the new limit form as \(x \to 0^+\). Since \(\ln x \to -\infty\), \((\ln x)^2 \to +\infty\), and \(\frac{1}{x} \to +\infty\), the limit becomes an indeterminate form \(\frac{\infty}{\infty}\), suitable for applying L'Hôpital's Rule.
Apply L'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\). The derivative of the numerator \((\ln x)^2\) is \(2 \ln x \cdot \frac{1}{x}\), and the derivative of the denominator \(\frac{1}{x}\) is \(-\frac{1}{x^2}\).
Simplify the resulting expression after differentiation and analyze the new limit as \(x \to 0^+\). This process will help determine the behavior of the original limit.

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주요 개념

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

Limits Involving Indeterminate Forms

Limits that result in expressions like 0·∞ or ∞·0 are called indeterminate forms. These require algebraic manipulation or special techniques to evaluate, as direct substitution does not yield a clear answer.
추천 영상:
07:01
Integrals Involving Natural Logs: Substitution

Behavior of the Natural Logarithm Near Zero

As x approaches 0 from the positive side, ln(x) approaches negative infinity. Understanding this behavior helps analyze expressions like x(ln x)², where the logarithm term grows without bound in magnitude.
추천 영상:
05:18
Derivative of the Natural Logarithmic Function

Techniques for Evaluating Limits (e.g., Substitution and L'Hôpital's Rule)

To evaluate limits involving indeterminate forms, techniques such as rewriting the expression, substitution, or applying L'Hôpital's Rule can be used. These methods transform the limit into a determinate form that can be evaluated directly.
추천 영상:
05:21
Finding Limits by Direct Substitution
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