Understanding the Limit of 1 Divided by Negative Infinity

To understand why 1-inftyfrac{1}{-infty} equals 0, we can look at the concept of limits in calculus.

Understanding Infinity

In mathematics, infty is not a number but a concept that represents an unbounded quantity. Similarly, -infty represents values that are decreasing without bound.

Division by Large Numbers

When you divide a fixed number like 1 by a very large negative number, the result approaches zero. For example:

1-10frac{1}{-10} -0.1 1-100frac{1}{-100} -0.01 1-1000frac{1}{-1000} -0.001

As the denominator becomes larger in magnitude (more negative), the result gets closer to zero.

Limit Approach

We can formalize this approach with limits. We can express this as:

limx#x2192;-inftyx→0lim_{x to -infty} frac{1}{x} 0

As x approaches negative infinity, 1x approaches zero.

Conclusion

Therefore, 1-infty 0 because as the denominator grows larger in magnitude (more negative), the overall fraction approaches zero.

Beware of Naive Extensions of Arithmetic Beyond Numbers

Infinity is not strictly a number, and dividing by it in arithmetic is not valid. However, when working with limits, you encounter expressions that look like division by infinity. This is what you should keep in mind.

Key Concepts:

infty is a concept, not a number. As the denominator becomes larger in magnitude, the fraction approaches zero. The limit of -1/n as n increases indefinitely is zero. In mathematical analysis, for any positive number x, there exists a number m such that the difference between -1/n and zero is less than x for all integers n greater than m.

Mathematical Analysis and Abstract Algebra

If you want to learn more about infinity, I recommend reading about mathematical analysis and abstract algebra, which are fundamental topics in a mathematics degree.