Evaluating Limits Involving Indeterminate Forms: A Comprehensive Guide

When dealing with limits involving expressions like 1^∞, it is essential to understand the intricacies of these indeterminate forms and the appropriate methods to evaluate them. This article provides a detailed explanation of a common misconception and the proper approach to solve such problems, highlighting key mathematical concepts and theorems.

Introduction to Indeterminate Forms

An indeterminate form, such as 1^∞, presents a challenge in limit evaluation because the conventional rules of arithmetic do not apply straightforwardly. The limit lim_{x→∞} (1 2/x)^x falls into this category, and it is crucial to comprehend the correct method to evaluate such limits.

Common Misconception: The Limit Evaluating to 1

A common misconception is that the expression 1 2/x approaches 1 as x approaches infinity, making the entire expression 1^∞ simply equal to 1. However, this is incorrect. The additional part, which is approximately 2/x, when multiplied by itself infinitely, leads to a non-trivial result.

Correct Method Using Direct Formula

The correct approach is to use a direct formula for evaluating such limits. The general formula is:

lim_{x→∞} (1 m/x)^(nx) e^(mn)

This formula can be derived from the definition of the number e, and it is particularly useful in simplifying the evaluation of certain indeterminate forms.

Applying the Direct Formula

Given the limit:

lim_{x→∞} (1 2/x)^x

We can directly apply the formula:

lim_{x→∞} (1 2/x)^(x/1) e^(2)

This simplifies the process significantly, and the limit evaluates to e^2, a special case of Euler’s number.

Classical Method Using L'H?pital's Rule

Though the direct formula is a more efficient method, it is also instructive to apply the classical approach using L'H?pital's Rule for educational purposes. Here’s how:

Step 1: Introducing the Natural Logarithm

We start with the natural logarithm to convert the limit into a more manageable form:

ln(L) ln(lim_{x→∞} (1 2/x)^x)

Since the natural logarithm is continuous, we can move the limit inside:

ln(L) lim_{x→∞} ln((1 2/x)^x)

This simplifies further to:

ln(L) lim_{x→∞} x ln(1 2/x)

Step 2: Applying L'H?pital's Rule

Both the numerator x ln(1 2/x) and the denominator 1/x approach 0 as x → ∞. Thus, we apply L'H?pital's Rule:

ln(L) lim_{x→∞} -[2/(x(1 2/x))] / [-1/x^2]

Simplifying the expression:

ln(L) lim_{x→∞} 2 / (1 2/x)

Multiplying both the numerator and the denominator by x:

ln(L) lim_{x→∞} 2x / (x 2)

We can split the fraction:

ln(L) lim_{x→∞} (2 - 4/x)

Finally, as x → ∞, the term -4/x approaches 0:

ln(L) 2

Therefore, L e^2.

Conclusion

The limit lim_{x→∞} (1 2/x)^x evaluates to e^2. Understanding the nuances of indeterminate forms, such as 1^∞, and applying the correct methods, such as the direct formula or L'H?pital's Rule, is crucial for accurate limit evaluation. This article provides a comprehensive guide to help you master these concepts.