Limits and Orders of Infinities

Think about something. Take a few values of and , say, .
You instantly notice that grows much faster than .

If we take infinitely larger values for each, goes up to infinity waaaaaaaaaaaay faster than just .

We say that * is a higher order of infinity than x.

Some elementary functions grow faster than others.

Take for example any form of x with no exponent, 2x, 3x, 100x, 1000x.

A positive power of x larger than 1 will inevitably reach a point where it will grow faster than any of what we just mentioned.

If we put them to a race, our newest competitor, any number raised to the x-th power, so, a constant number c of your choice, in the form will reach a point where it will grow incredibly faster than any power of x. X is now in the exponent.

And finally, the factorial function, which we said is the fastest growing elementary function, as it goes towards infinity, it will take the infinitely many numbers behind it and multiply them all with itself, being the fastest growing function.