Determine the following limits.
lim w→∞ (ln w²) / (ln w³ + 1)

Verified step by step guidance

1. Identify the limit expression: \( \lim_{w \to \infty} \frac{\ln w^2}{\ln w^3 + 1} \).
2. Simplify the expression: \( \ln w^2 = 2 \ln w \) and \( \ln w^3 = 3 \ln w \).
3. Rewrite the limit using the simplified expressions: \( \lim_{w \to \infty} \frac{2 \ln w}{3 \ln w + 1} \).
4. Factor out \( \ln w \) from the denominator: \( \lim_{w \to \infty} \frac{2 \ln w}{\ln w (3 + \frac{1}{\ln w})} \).
5. Cancel \( \ln w \) and evaluate the limit: \( \lim_{w \to \infty} \frac{2}{3 + \frac{1}{\ln w}} \).

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of a function as w approaches infinity. Understanding limits helps in analyzing the asymptotic behavior of functions and is crucial for evaluating expressions that may not be directly computable.

Natural Logarithm

The natural logarithm, denoted as ln, 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. In the given limit, the natural logarithm of w raised to a power is involved, which simplifies to a multiplication of the exponent and ln(w), illustrating the properties of logarithms.

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 ∞/∞. 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. This rule is particularly useful in the context of the given limit, as it can simplify the evaluation of the logarithmic expressions.