60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 
lim_x→∞ ln ((x +1) / (x-1))

Verified step by step guidance

1. Identify the form of the limit as x approaches infinity: lim_(x→∞) ln((x + 1) / (x - 1)). This is an indeterminate form of type ∞/∞.
2. Apply l'Hôpital's Rule, which is used to evaluate limits of indeterminate forms. First, differentiate the numerator and the denominator of the argument of the logarithm separately.
3. Differentiate the numerator (x + 1) to get 1, and the denominator (x - 1) to get 1. The limit now becomes lim_(x→∞) ln(1).
4. Since ln(1) is a constant, the limit simplifies to 0.
5. Conclude that the limit of the original expression as x approaches infinity is 0.

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 at points where it may not be explicitly defined, such as at infinity or discontinuities. Evaluating limits is crucial for determining the continuity and differentiability of functions.

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 key function in calculus, particularly in growth and decay problems, and is often used in limits involving exponential functions. Understanding its properties, such as its domain and range, is essential for evaluating limits that involve logarithmic expressions.

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, especially when dealing with complex functions.