Question

Indeterminate Powers and ProductsFind the limits in Exercises 53–68.66. lim (x → 0⁺) x (ln x)²

Accepted Answer

Identify the form of the limit as $$x 	o 0^+$$ 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}}$$. Check the new limit form as x$$ 	o 0^+$$. Since $$\ln x 	o -\infty$$, $$(\ln $$x)^2$$ 	o +\infty$$, and $$\frac{1}{x} 	o +\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 $$x 	o 0^+. $$This process will help determine the behavior of the original limit.

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

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Concetti chiave

Limits Involving Indeterminate Forms

Behavior of the Natural Logarithm Near Zero

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