Understanding The Little o
Let's take the previous functions again: We'll do the same thing as last lesson, but now, instead of going to infinity, let's use infinitesimals.
Instead of going all the way up, go all the way down to zero. From the right.
Now, take all numbers between zero and 1. So, all numbers that look like : .svg)
Now, try squaring them: Indeed, positive increasing powers of numbers between 0 and 1, 0 and 1 not included, make themselves smaller instead of bigger.
Notice a familiar pattern? We have the same thing, except, is now the one that's negligible.
Calculate the limit again, but for x approaching zero (obviously, from the right): Answer:
"wtf do you mean?"
Now, simply flip the fraction on that limit. See what you get.
See it now?
being negligible to , or, as x approached a point c meant that is practically zero, or "nothing" compared to when going to a certain point c.
The same way, being negligible to , or, as x approached a point c meant that this is also zero, or nothing compared to when going to a certain point c.
When we had the case that was negligible to , this meant that the limit of the ratio of over was zero, and that the limit of the ratio of over tended to infinity.
The same way, when we had the case that was negligible to , this meant that the limit of the ratio of over as x approached a point was zero, and that the limit of the ratio of over tended to infinity.
PAY ATTENTION TO SIGNS IN CASES OF INFINITIES!
Convention: Little-o in equations
In Mathematical Analysis, at least, the Politecnico di Torino course, by convention we will interpret written in an equation to mean "x plus something negligible to x", in the case of
Similarly, if you see where is any function in terms of x, for example , you are to interpret that as x plus a function of x, for example , where this function of x, or our example, is negligible to x.
This will usually be seen when approaching a point or when analyzing a function's behaviour tending to infinities, zeroes, or points where they're undefined.
Next, we'll talk about common cases where this limit exists and is neither zero or infinite. So, a finite limit that exists in .