17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0⁺ (1 - ln x) / (1 + ln x)

Verified step by step guidance

1. First, identify the form of the limit as x approaches 0 from the positive side. Substitute x = 0⁺ into the expression (1 - ln x) / (1 + ln x) to check if it results in an indeterminate form like 0/0 or ∞/∞.
2. Since ln(x) approaches -∞ as x approaches 0⁺, both the numerator (1 - ln x) and the denominator (1 + ln x) approach ∞, resulting in the indeterminate form ∞/∞. This indicates that l'Hôpital's Rule can be applied.
3. Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, then the limit can be evaluated as the limit of f'(x)/g'(x) as x approaches the same point, provided the derivatives exist.
4. Differentiate the numerator: The derivative of 1 - ln x with respect to x is -1/x.
5. Differentiate the denominator: The derivative of 1 + ln x with respect to x is 1/x.

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 where the function may not be defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It 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 rule simplifies the process of finding limits in complex functions.

Natural Logarithm (ln)

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in limits and derivatives, due to its unique properties, such as ln(1) = 0 and its derivative being 1/x. Understanding the behavior of ln(x) as x approaches 0 is essential for evaluating limits involving this function.