Understanding the Indeterminate Form 1∞ in Mathematics

Understanding the Indeterminate Form 1∞ in Mathematics

The expression 1^{infty} is considered an indeterminate form in mathematics. This means that its value is not well-defined without additional context. This article will delve into why this is the case, how it arises in calculus, and provide examples to illustrate its behavior.

What is the Indeterminate Form 1∞?

The expression 1^{infty} can be interpreted in different ways, leading to various outcomes. The base 1 raised to any finite power will always yield 1. However, when dealing with limits in calculus, the indeterminate form can lead to different values. For instance, when a function approaches 1 and another approaches infinity, the overall limit can vary based on their rates of change.

Base of 1

When you raise 1 to any power, the result is always 1. So 1^x 1 for any finite x. This is straightforward and intuitively clear. However, in calculus, things can get more complex.

Limit Context in Calculus

The indeterminate form 1∞ arises particularly in the context of limits. Consider the expression lim_{x to a} f(x)^{g(x)} where f(x) to 1 and g(x) to infty. The overall limit can vary depending on how f(x) approaches 1 and how g(x) approaches infinity. Here are a couple of examples to illustrate this:

Example 1

If f(x) 1 frac{1}{n} and g(x) n as n to infty, then (1 frac{1}{n})^n approaches the value of e.

Example 2

Conversely, if f(x) 1 - frac{1}{n} and g(x) n, then (1 - frac{1}{n})^n approaches 1/e.

In summary, 1∞ does not have a single value and is context-dependent. Often, resolving the specific outcome requires the use of limits.

Proof by Induction

To formally prove that 1^n 1 by induction, let's follow these steps:

Hypothesis

The hypothesis is that 1^n 1. We will prove this for all positive integers n.

Base Case

For n 1, we have 1^1 1. This is true by definition.

Inductive Step

Assume the hypothesis is true for n m. That is, 1^m 1.

Proof for n m 1

We need to prove that 1^{m 1} 1.

LHS  1^{m 1}  1 * 1^m

By the inductive hypothesis, we know that 1^m 1. Thus,

LHS  1 * 1  1  RHS

By taking the limit as n to infty on both sides, we get our desired result. Hence, the hypothesis is true for all n.

This proof by induction confirms that 1 raised to any positive integer power is always 1.

Conclusion

The expression 1∞ is indeterminate because its value depends on the context, such as the rates at which the functions in the limit approach 1 and infinity. Understanding these nuances is crucial in calculus and other advanced mathematical applications.