Evaluating Limits Involving Factorials: A Comprehensive Guide

When faced with the challenge of evaluating limits that involve factorials, it is essential to have a systematic approach. This article provides a step-by-step methodology, including the use of Stirling's approximation and L'H?pital's rule, to simplify and evaluate such limits. Whether you are a student, a mathematician, or a professional in need of practical solutions, this guide will help you navigate through complex expressions.

Identifying the Limit

The first step in evaluating a limit involving factorials is to clearly identify the limit you need to evaluate. The general form of such a limit might look like this:

[lim_{n to infty} frac{n!}{a^n}] where a is a constant and n! represents the factorial of n.

Applying Stirling's Approximation

For large values of n, the factorial can be significantly simplified using Stirling's approximation. This mathematical tool provides an approximation of the factorial function:

[n! sim sqrt{2 pi n} left(frac{n}{e}right)^n]

This approximation is particularly useful when dealing with limits as n approaches infinity, as it helps in reducing the complexity of the expression.

Simplifying the Expression

Once Stirling's approximation is applied, the next step is to substitute it into the original limit and simplify the expression. For example, consider the limit:

[lim_{n to infty} frac{n!}{a^n}]

After applying Stirling's approximation, this simplifies to:

[lim_{n to infty} frac{sqrt{2 pi n} left(frac{n}{e}right)^n}{a^n}]

This further simplifies to:

[lim_{n to infty} sqrt{2 pi n} left(frac{n}{ae}right)^n]

Analyzing the Growth Rates

The final step involves analyzing the behavior of the terms as ntoinfty. The key is to determine how the terms grow relative to each other. For instance, in the previous example:

If (frac{n}{ae} 1), the limit diverges to infinity. If (frac{n}{ae} 1), the limit approaches 0.

In the provided example, where the limit is:

[lim_{n to infty} frac{n!}{n^n}]

Substituting Stirling's approximation yields:

[lim_{n to infty} frac{sqrt{2 pi n} left(frac{n}{e}right)^n}{n^n}]

This simplifies to:

[lim_{n to infty} sqrt{2 pi n} left(frac{1}{e}right)^n]

Here, since (frac{1}{e} 1), the term (left(frac{1}{e}right)^n) approaches 0 much faster than (sqrt{2 pi n}) grows, resulting in:

[lim_{n to infty} sqrt{2 pi n} left(frac{1}{e}right)^n 0]

Therefore, the limit is:

[lim_{n to infty} frac{n!}{n^n} 0]

Using L'H?pital's Rule if Necessary

In some cases, you might encounter an indeterminate form such as (frac{infty}{infty}) or (frac{0}{0}). In such situations, L'H?pital's rule can be applied to differentiate the numerator and the denominator until the limit can be evaluated. This rule is particularly useful when the standard methods of analysis are insufficient.

Conclusion

Evaluating limits involving factorials requires a systematic approach, combining the use of Stirling's approximation, simplification, and analysis of growth rates. For more complex indeterminate forms, L'H?pital's rule can be a valuable tool. By following these steps, you can tackle even the most challenging problems involving factorials and limits.