As a domain expert in mathematics, I'm often asked about the concept of infinity and its relationship with numbers. Infinity is not a number but rather an abstract concept that represents an unbounded quantity. It's a useful tool in calculus and other areas of mathematics to describe things that are larger than any finite number. Let's delve into the question of what infinity over a number equals, considering the different scenarios that can arise.

Step 1: Understanding Infinity

When we speak of infinity divided by a number, we are essentially discussing a limit process. In calculus, limits are a way to understand what happens to a function as its input approaches a certain value. When we say "infinity over a number," we're often referring to the limit of a function as its denominator approaches a finite number, not that infinity is literally being divided.

Infinity Over a Non-Zero Number

When infinity is divided by a non-zero number, the result is still infinity. This is because as the denominator approaches a non-zero finite value, the fraction approaches infinity. Mathematically, this can be expressed as:

\[
\lim_{x \to a} \frac{\infty}{x} = \infty \quad \text{for} \quad x \neq 0
\]

Here, \( a \) is any non-zero finite number, and \( \lim \) denotes the limit as \( x \) approaches \( a \).

Infinity Over Zero

If we were to consider infinity over zero, we encounter an indeterminate form. Division by zero is undefined in mathematics, and infinity divided by zero does not yield a meaningful result. It's not infinity, nor is it a finite number. It's simply undefined.

A Number Over Infinity

Conversely, when a finite number is divided by infinity, the result is zero. This is because as the denominator grows without bound, the fraction approaches zero. The mathematical expression for this is:

\[
\lim_{x \to \infty} \frac{a}{x} = 0 \quad \text{for} \quad a \neq 0
\]

Here, \( a \) is any finite number.

**Indeterminate Forms and L'Hôpital's Rule**

When we encounter expressions like \( 0/0 \) or \( \infty/\infty \), we have indeterminate forms. These forms do not have a clear value and require further manipulation to determine a limit. One common method to deal with these is L'Hôpital's Rule, which states that if the limit of a function results in an indeterminate form \( 0/0 \) or \( \infty/\infty \), then this limit is equal to the limit of the derivatives of the numerator and denominator, provided this new limit exists.

\[
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{if} \quad \lim_{x \to a} \frac{f(x)}{g(x)} \text{ is } 0/0 \text{ or } \infty/\infty
\]

Summary

- Infinity over a non-zero number is infinity.
- Infinity over zero is undefined.
- A number over infinity is zero.
- Indeterminate forms like \( 0/0 \) or \( \infty/\infty \) may be resolved using L'Hôpital's Rule if applicable.

Now, let's proceed to the next step.

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A number, you're done. A number over zero or infinity over zero, the answer is infinity. A number over infinity, the answer is zero. 0/0 or --/--, use L'H?pital's Rule.read more >>