60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 
lim_ x→0 ⁺ | ln x | ˣ

Verified step by step guidance

1. Recognize that as x approaches 0 from the right, ln(x) approaches negative infinity, making the expression |ln(x)| approach positive infinity.
2. Rewrite the limit as lim_{x→0⁺} |ln(x)|^x, which can be expressed as lim_{x→0⁺} (ln(x))^x since |ln(x)| is positive for x > 0.
3. Take the natural logarithm of the limit to simplify the expression: let L = lim_{x→0⁺} (ln(x))^x, then ln(L) = lim_{x→0⁺} x * ln(ln(x)).
4. Evaluate the limit ln(L) = lim_{x→0⁺} x * ln(ln(x)). As x approaches 0, ln(ln(x)) approaches negative infinity, and x approaches 0, creating an indeterminate form of 0 * (-∞).
5. Apply l'Hôpital's Rule to resolve the indeterminate form by rewriting it as a fraction: ln(ln(x)) / (1/x) and differentiate the numerator and denominator before re-evaluating the limit.