Principal Parts With Respect To Test Infinite(simal) Functions

You can determine a "relative" magnitude, or rate at which a function that is infinite or infinitesimal tending to a point c, is approaching this infinity or zero in comparison to another function.

The function you'll use is called the "test function". Sometimes you may be given one, sometimes you can just take whatever you want, as long as it's also an infinite or infinitesimal according to the function you're testing.

If you're not given a test function to use, you may use:

Suppose that as x tends to c, is controlled by our test function . So, this limit exists and is finite and non-zero. Then, it's of the same order.

But, what if it's controlled, instead, by a power of this test function?
We say that our function is an infinite/simal of order with respect to our test function

Since this limit exists and is finite, we, of course, have a value l so that:
Basically just the value of this limit that exists.

"So then wtf is a principal part?"

Well, these functions may not be related, like, at all. They could be totally different.
But because of this existing limit, as we tend to this point, we have a relation between these functions. The ratio of these functions and our or the power of our , --

As we get to this point, it becomes possible to get from that test function to a very close version of our original function we're testing, called the principal part.

"But, why not just the original? "

Because sometimes this could help us in approximations or finding what has the most impact on this particular function as we tend to a point. Or sometimes we just can't.

So how would you? Well, we did find that since it's controlled, it means that it will behave as a ratio of our function over a power of this test function of ours.

So, the first operation you do is raise that test function to such power.

We also did find that there existed a limit l in this case, which is the ratio of our function over this power of our test function.

The second operation you do, therefore, is multiply this power by this value l of said limit. You have now determined an estimate or another version of your original function, in terms of your test function. That's what the principal part is.

You are done.

Now things will start to ramp up a bit.

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