The Big O

So, we've learned that, in a nutshell, the o-notation, landau notation, whatever you wanna call it, is a way of relating functions based on how they approach a point.

Or rather, the limit of the ratio of these two functions.

But, of course, there will be infinite scenarios where this limit exists, but is neither infinity nor zero.
"What happens now?"
Now? Let's go practice medicine!

The Big O

Written as , or, , the Big O is read as "is controlled by", and is used when we have said limit to at a point, but it's finite and non-zero.

From this, you immediately understand that the Little-o is a sub-case of the Big-O. A sub-case means "a particular version of something". And yeah, this limit in this case for is zero!

Let's now consider some other example functions. Calculate the limit as x approaches to 5 of over : This will work for any point, not just the number 5 in this case, because, the fraction simplifies, losing dependence on the x variable:
The limit of over will always be no matter what x we choose. In a sense, the way is, will "force" f(x) into a certain limit.

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