What is Infinity divided by Infinity?

Answer 1

It is indeterminate. There are many other inderterminate forms. It is not at all the same as 3/3 for example. You can see this with limits and some calculus rules. You must apply the L'Hospital theorem by deriving the numerator and the denominator of the equation that gave you infinity over infinity.

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Why ∞/∞ is not 1

One could think that ∞/∞ = 1, but this is wrong.

The answer depends on the kind of infinity: in fact, there are different kinds of infinity.

For example, consider f(x) = x2 and g(x) = x. In the limit x→∞ of the function f(x)/g(x), we have

limx→∞ f(x)/g(x) = limx→∞ x2/x = limx→∞ x = ∞;

so, both f(x) and g(x), in that limit, equal infinity, but f(x)/g(x) ≠ 1. If we have f(x) = 2x and g(x) = x, both f(x) and g(x) equal infinity (for x→∞), but

limx→∞ f(x)/g(x) = limx→∞ 2x/x = limx→∞ 2 = 2 ≠ 1.

So you see that infinity is something to check everytime!

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Addition: Since infinity is not a set number, you cannot assume that infinity divided by infinity would equal one. Infinity is an indeterminate number.
1

To touch on this whatever you take and divide by the same number will always give you one.

2

Infinity divided by infinity is not equal to 1, But it is undefined, not another infinity. This would help you:
First, I am going to define this axiom (assumption) that infinity divided by infinity is equal to one:

∞
-
∞
= 1

Since ∞ = ∞ + ∞, then we are going to substitute the first infinity in our axiom:

∞ + ∞
---
∞
= 1

The next step is to split this fraction into two fractions:

∞
-
∞
+ ∞
-
∞
= 1

Next, substitute the axiom twice into the equation, we get:

1 + 1 = 1

Finally, this can be rewritten as:

2 = 1

Therefore, infinity divided by infinity is NOT equal to one. Instead we can get any real number to equal to one when we assume infinity divided by infinity is equal to one, so infinity divided by infinity is undefined.