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

Verified step by step guidance

1. 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 \(0 \cdot \infty\).
2. 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 \(\frac{(\ln x)^2}{\frac{1}{x}}\).
3. 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.
4. 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 \(-\frac{1}{x^2}\).
5. 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.

Key Concepts

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.

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.

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.