Question

Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. $$x^{10} \ln ^{10} x ; x^{11}$$

Short answer

Answer: The function \(x^{11}\) grows faster than the function \(x^{10}\ln^{10}x\).

Step-by-step solution

Find the limit

In this step, we want to find the limit of the ratio of the two given functions as \(x\) approaches infinity: $$\lim_{x\to\infty}\frac{x^{10}\ln^{10}x}{x^{11}}$$

Simplify the expression

We can simplify the expression inside the limit by dividing both terms in the numerator and the denominator by \(x^{10}\): $$\lim_{x\to\infty}\frac{x^{10}\ln^{10}x}{x^{11}}=\lim_{x\to\infty}\frac{\ln^{10}x}{x}$$

Apply L'Hopital's Rule

Since the limit is in the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) as \(x\to\infty\), we can use L'Hopital's Rule, which states that for functions \(f(x)\) and \(g(x)\): $$\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}\quad \text{if the limit on the right exists}$$ We will differentiate the numerator and denominator with respect to \(x\): $$\frac{d}{dx}(\ln^{10}x) = 10\cdot \ln^9x \cdot\frac{1}{x}$$ $$\frac{d}{dx}(x) = 1$$ Now substituting these into the limit: $$\lim_{x\to\infty}\frac{10\cdot \ln^9x \cdot\frac{1}{x}}{1}$$

Simplify the expression again

We can simplify the expression inside the limit by removing the constant 10 from the limit: $$\lim_{x\to\infty}\frac{10\cdot \ln^9x \cdot\frac{1}{x}}{1}=10\cdot\lim_{x\to\infty}\frac{ \ln^9x }{x}$$

Evaluate the limit

Since \(\ln x\) grows slower than any power of \(x\), the limit of the ratio \(\frac{ \ln^9x }{x}\) as \(x\to\infty\) is 0: $$10\cdot\lim_{x\to\infty}\frac{ \ln^9x }{x}=10\cdot 0=0$$

Conclusion

The final limit is 0, which means that the first function, \(x^{10}\ln^{10}x\), grows slower than the second function, \(x^{11}\). Therefore, the function \(x^{11}\) grows faster than the function \(x^{10}\ln^{10}x\).

Key concepts

Limit of a Function

Understanding the limit of a function is crucial when analyzing the behavior of functions as the input values become very large or very small.

The limit of a function as the variable approaches a certain value is the outcome that the function's outputs get progressively closer to, as the variable comes ever closer to that specific value. This does not necessarily mean that the function will ever reach that outcome exactly, but rather that it can get arbitrarily close to it.

In mathematical terms, the limit of a function as the variable, say \( x \), approaches a certain value \( a \) is represented by \( \lim_{x \to a} f(x) \). If \( x \) approaches infinity, the function is said to have an infinite limit, and we examine the end behavior of the function—essentially, what happens as \( x \) grows without bound.

L'Hopital's Rule

When evaluating limits, we sometimes encounter indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms don't provide clear answers at a glance, and that's where L'Hopital's Rule steps in as a powerful tool.

L'Hopital's Rule states that if we have a limit of the form \( \lim_{x \to c} \frac{f(x)}{g(x)} \) that results in an indeterminate form, then under certain conditions, this limit can be evaluated by finding the limit of the derivatives of the numerator and the denominator. This is expressed as:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

provided that the limit on the right-hand side exists and does not also result in an indeterminate form. This rule is a consequence of the mean value theorem and generally applies to cases where \( f \) and \( g \) are differentiable near \( c \) and \( g'(x) \) is not zero.

Indeterminate Forms

Within the realm of calculus, indeterminate forms arise when evaluating limits that cannot be determined from the information given. These forms include \( 0/0 \), \( \infty/\infty \), \( 0 \cdot \infty \), \( \infty - \infty \), \( 1^\infty \), \( 0^0 \), and \( \infty^0 \).

Though they may initially seem nonsensical or undefined, indeterminate forms tell us that further analysis is needed to properly evaluate the limit. One common method to resolve indeterminate forms, particularly \( 0/0 \) or \( \infty/\infty \), is to use L'Hopital's Rule, as illustrated in the step-by-step solution of the original exercise.

Infinite Limits

An infinite limit occurs when the value of a function increases or decreases without bound as the variable approaches a certain point or as it approaches positive or negative infinity.

In mathematical notation, we write \( \lim_{x \to a} f(x) = \infty \) if the function grows without bound as \( x \) approaches the value \( a \). Conversely, if the function decreases without bound, we write \( \lim_{x \to a} f(x) = -\infty \). When considering limits at infinity, such as \( \lim_{x \to \infty} f(x) \), we are interested in the end behavior of \( f(x) \) as \( x \) grows larger and larger.

The study of infinite limits is important for understanding asymptotic behaviors—how the function behaves near a vertical asymptote or as it goes off to infinity horizontally. It can provide insights into the growth rate comparison of different functions, especially when determining the long-term behavior and dominance among multiple functions.