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Chapter 4: Problem 74

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates. $$x^{2} \ln x ; x^{3}$$

Short Answer

Expert verified
Answer: The function \(x^3\) grows faster than the function \(x^2\ln x\).

Step by step solution

01

Set up the limit

To compare the growth rates of the given functions, we will find the limit of their ratio as x approaches infinity: $$\lim_{x\to\infty} \frac{x^2\ln x}{x^3}$$
02

Simplify the limit

We can simplify the limit by canceling a common term, \(x^2\), from both the numerator and the denominator: $$\lim_{x\to\infty} \frac{\ln x}{x}$$
03

Apply L'H么pital's Rule

Since the limit is in the indeterminate form \(\frac{\infty}{\infty}\), we can apply L'H么pital's Rule by differentiating both the numerator and the denominator with respect to x: $$\lim_{x\to\infty} \frac{\frac{d}{dx}(\ln x)}{\frac{d}{dx}(x)} = \lim_{x\to\infty} \frac{1/x}{1}$$
04

Evaluate the limit

Evaluating the limit, we get: $$\lim_{x\to\infty} \frac{1/x}{1} = 0$$
05

Interpret the result

Since the limit of the ratio of the functions is 0 as x approaches infinity, it means that the function in the denominator, \(x^3\), grows faster than the function in the numerator, \(x^2\ln x\). As a result, we can conclude that the function \(x^3\) grows faster than the function \(x^2\ln x\).

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