What is Infinity Divided by Zero?

Infinity divided by zero is undefined.

Here's a breakdown of why:

• Infinity is Not a Number: It's crucial to understand that infinity (∞) isn't a real number. It represents a concept of something without any bound or limit. Arithmetic operations like division are generally defined for numbers, so directly dividing infinity by zero (or any number) doesn't have a standard mathematical meaning.

• Division by Zero is Undefined: Division by zero is undefined in standard arithmetic. When we say "a / b = c," it means "b * c = a." If b = 0, then "0 * c = a." The only value of 'a' that can satisfy this for all 'c' is 0. For any non-zero 'a,' there's no solution for 'c,' hence the operation is undefined.

• Limits and Approaching Infinity/Zero: While direct division is undefined, we can use limits to examine what happens when a quantity approaches infinity and is divided by a quantity approaching zero. This involves calculus.

  • Example 1: (Approaching Infinity) / (Approaching Zero):

    Consider the limit: lim (x→∞) x / (1/x)

    As x approaches infinity, x also approaches infinity, and 1/x approaches zero.
    We can simplify the expression to: lim (x→∞) x².
    As x gets infinitely large, x² also gets infinitely large. Therefore, the limit is infinity.

  • Example 2: (Approaching Infinity) / (Approaching Zero):

    Consider the limit: lim (x→∞) 5x / (1/x²)

    As x approaches infinity, 5x also approaches infinity, and 1/x² approaches zero.
    We can simplify the expression to: lim (x→∞) 5x³.
    As x gets infinitely large, 5x³ also gets infinitely large. Therefore, the limit is infinity.

  • Indeterminate Forms: Expressions like ∞/∞, 0/0, ∞ * 0, ∞ - ∞, 1^∞, 0⁰, and ∞⁰ are called "indeterminate forms." This means that without further analysis, you can't determine the limit's value. The limit may be a finite number, infinity, negative infinity, or it might not exist at all. The value depends on the specific functions involved.

• L'Hôpital's Rule: When dealing with limits that result in indeterminate forms like 0/0 or ∞/∞, L'Hôpital's Rule is often applied. This rule states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, and if f'(x) and g'(x) exist (and g'(x) is not zero), then the limit of f(x)/g(x) is equal to the limit of f'(x)/g'(x). This can often simplify the expression and allow you to evaluate the limit.

In summary, while you cannot directly calculate infinity divided by zero, you can use the concept of limits to analyze what happens when a function approaches infinity and is divided by a function that approaches zero. The result can be infinity, a finite number, or may not exist depending on the specifics of the functions involved.