Landau Notation and Local Comparison of Functions - The little o.

Yeah, this one sucks.

Landau Notation is, in a nutshell, used to relate how fast functions grow or approach a point related to each-other.

You may also have heard of this as the "Little-O/Big-O" or "Theta-Omega" notation.

Definition and Prelude to Rates of Growth or Approach

As you may have seen on the lecture 10 file, on the definition, we first defined two functions defined in a neighbourhood around a point c.

But why this "neighbourhood"? Because we need limits.
The limit is as a function tends to a point, and, you know, to approach a point, the function needs to be defined around it. That's why we need a neighbourhood. Because we'll use limits.

The Little O:

This may seem a bit confusing, but it's just so you can understand what we mean when we say, for example, that is as we approach a point. Here it is!

"I still don't get it!" -- Yeah, me neither. Let's try again!

Let's take these two functions as an example: center

Let's go to infinity.

The function will inevitably reach a point where it will be unimaginably large. Powers!
It will be so large that, you know, if we compare to it, x is practically nothing: As , is negligible compared to :

And for the final infinity stone ~~ha, ha~~, based on the above info, calculate this limit: We already established that as we go to infinity, this is so unimaginably large that f(x) is zero compared to it. Here's the graph, including the ratio , in orange, in case you are confused:
250|center

You may still be a bit confused, but that's okay. You'll get it now.
Understanding The Little o