Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x² ln x; x³

Verified step by step guidance

1. First, understand that to compare the growth rates of two functions, we can use the concept of limits. Specifically, we can evaluate the limit of the ratio of the two functions as x approaches infinity.
2. Set up the limit expression: \( \lim_{x \to \infty} \frac{x^2 \ln x}{x^3} \). This will help us determine which function grows faster.
3. Simplify the expression inside the limit: \( \frac{x^2 \ln x}{x^3} = \frac{\ln x}{x} \). This simplification is achieved by canceling \( x^2 \) from both the numerator and the denominator.
4. Evaluate the limit \( \lim_{x \to \infty} \frac{\ln x}{x} \). As x approaches infinity, \( \ln x \) grows slower than any polynomial function, and \( x \) grows faster, leading the limit to approach 0.
5. Conclude that since the limit is 0, \( x^3 \) grows faster than \( x^2 \ln x \) as x approaches infinity.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for analyzing the behavior of functions, especially as they tend toward infinity or a specific value. In this context, limits help determine the growth rates of the functions x² ln x and x³ as x becomes very large.

Growth Rates

Growth rates describe how a function increases as its input increases. In calculus, we often compare functions to see which grows faster by examining their limits. This comparison can be made using techniques like L'Hôpital's Rule or by analyzing the leading terms of the functions involved, which is crucial for understanding the relative growth of x² ln x and x³.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in comparing the growth rates of functions like x² ln x and x³, allowing for a clearer analysis of their behavior as x approaches infinity.