Exploring the Limitations of the Expression xy / (x - y) for Distinct Positive Real Numbers

When working with mathematical expressions involving distinct positive real numbers, understanding the behavior of a function is crucial. In this article, we delve into the expression xy / (x - y) and explore its limitations and the implications of its behavior.

Introduction

Given the expression xy / (x - y), where x and y are distinct positive real numbers, we aim to investigate the minimum value this expression can achieve. To start, let's assume a specific case where y 1.

Case Analysis: y 1

Let's substitute y 1 into the expression, giving us:

xy / (x - 1) x / (x - 1)

Behavior as x Approaches 1 from the Right

Let's examine the behavior of the expression x / (x - 1) as x gets closer to 1 from the right. When x is slightly greater than 1, the expression becomes:

2.01 / -0.01 -201

Similarly, when x is even closer to 1, say 1.0001, the expression evaluates to:

2.0001 / -0.0001 -20001

As x approaches 1 from the right, the denominator x - 1 becomes a very small negative number, causing the expression to approach negative infinity. This demonstrates that the expression can be made arbitrarily negative, implying that there is no minimum value.

Conclusion for y 1

Based on the above analysis, we conclude that the minimum value of the expression xy / (x - y) does not exist when y 1. The expression can be made arbitrarily negative, and hence there is no finite minimum value.

General Case Analysis

However, to ensure a comprehensive understanding, let's consider the general case where x and y are distinct positive real numbers. We can use another approach to show that the minimum value does not exist.

Consider the expression xy / (x - y) as x and y are distinct positive real numbers. By setting y 1/n and x 1, the expression becomes:

xy / (x - y) 1 * (1/n) / (1 - 1/n) (2/n) / (-1/n) -2n

As n approaches infinity, the expression -2n approaches negative infinity. This further confirms that the minimum value of the expression does not exist.

Conclusion

In conclusion, the expression xy / (x - y) for distinct positive real numbers x and y does not have a minimum value. The expression can be made arbitrarily negative, and thus the minimum value is negative infinity (-oo).

Key Takeaways

The expression xy / (x - y) can be made arbitrarily negative for distinct positive real numbers x and y. The minimum value of the expression does not exist and approaches negative infinity. This behavior is observed both when y 1 and in the general case where y 1/n as n approaches infinity.

Related Keywords

Minimum value, distinct positive real numbers, limits