Landau Notation and Orders of Infinite(simal)s
Let's bring back that veeery first example that we saw when entering the Landau Notation chapter:Let's first see what happens when we make these functions infinitesimals. For this, x will tend to zero. Since we have that makes it a bit more complicated for a double-sided limit, we'll consider it as x tends to zero from the right, again.
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As we previously established, our , when we go towards zero, will be infinitely smaller in comparison to the , practically "nothing compared to it"
Given that this larger power is "negligible" as we tend to zero, so, have an infinitesimal, goes to zero much faster than , as our x goes to zero.
g(x) is an infinitesimal of higher order than f(x)
In infinitesimals, larger powers are negligible.
Now let's go to infinity again!
Now, as our x gets infinitely larger, obviously, is unimaginably large compared to . In other words, is negligible, can be considered as zero compared to this when we tend to infinity.
g(x) is an infinite of higher order than f(x)
In infinites, smaller powers are negligible.
For infinites, functions that approach infinity as we approach a point:
For infinitesimals, functions that approach zero as we approach a point:
Magnitude Parity
For functions that are infinites or infinitesimals, if one of them controls each-other, as in, the limit of their ratios exists, is finite and non-zero, but they both are infinites or infinitesimals as x tends to a point, they are infinite(simal)s of the same order.
Your suffering is almost over.
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