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. First, recognize that the limit involves an indeterminate form. As x approaches 0 from the right, ln(x) approaches negative infinity, and x approaches 0, creating the form 0^(-∞).
2. To handle this indeterminate form, consider rewriting the expression |ln(x)|^x as e^(x * ln(|ln(x)|)). This transformation allows us to work with the exponent separately.
3. Now, focus on evaluating the limit of the exponent: lim_(x→0⁺) x * ln(|ln(x)|). This is an indeterminate form of type 0 * (-∞), which can be addressed using l'Hôpital's Rule.
4. Apply l'Hôpital's Rule to the limit lim_(x→0⁺) x * ln(|ln(x)|). First, rewrite it as lim_(x→0⁺) ln(|ln(x)|) / (1/x), which is an ∞/∞ form suitable for l'Hôpital's Rule.
5. Differentiate the numerator and the denominator separately: the derivative of ln(|ln(x)|) with respect to x and the derivative of 1/x with respect to x. Then, evaluate the limit of the resulting expression as x approaches 0 from the right.

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or infinity. Evaluating limits is essential for defining derivatives and integrals, which are core components of calculus.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a crucial function in calculus, particularly in growth and decay problems, and is often involved in limits and derivatives. Understanding the properties of the natural logarithm, such as its behavior as x approaches 0, is vital for evaluating limits involving ln(x).

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This technique simplifies the evaluation of complex limits, making it a powerful tool in calculus.