Question

What's the difference between: undefined, does not exist and infinity?

Answer 1

You tend to see "undefined" when dividing by zero, because how can you separate a group of things into zero partitions? In other words, if you had a cookie, you know how to divide it into two parts---break it in half. You know how to divide it into one part---you do nothing. How would you divide it into no parts? It's undefined.

`1/0 = "undefined"`

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You tend to see "does not exist" when you encounter imaginary numbers in the context of real numbers, or perhaps when taking a limit at a point where you get a two-sided divergence, such as:

`lim_(x->0^+) 1/x = oo`
`lim_(x->0^-) 1/x = -oo`

Therefore:
`lim_(x->0) 1/x => "DNE"`

graph{1/x [-10, 10, -5, 5]}

This would be due to the fact that a limit does not exist when the limit from both the positive and negative direction differ (it's like trying to make two north poles of magnets meet, and when they meet, if they meet, that is their limit---but they never meet).

In those cases, either the limit from one side exists only, or the domain of the function does not contain the desired limit.

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Infinity is something that exists for us to quantify something that can never be truly reached in the absolute sense. Infinity is just an arbitrarily large number that we attribute to solutions that we know will keep increasing or decreasing forever.

For example...

`lim_(x->oo) x^2 = oo`

simply means we keep moving to the right and repeatedly determine the value of `x^2` at each arbitrary `x` value... forever. The "final" value is then called `oo`, even though we never actually reach a final value. But we want to reach one, so we called it infinity.