Infinitesimal functions

and

have the same order as , since

which is a finite number.

Example 2

Given infinitesimal functions

and ,

as , find the limit of their ratio.

Solution:

The limit equals zero. Therefore, is an infinitesimal function of the higher order of smallness with respect to .

Example 3

If and , then is an infinitesimal function of the third order with respect to as .

Really,

,

which is a finite number.

Example 4

Given infinitesimal functions and as , find the limit of their ratio.

Example 5

Both functions,

and ,

are infinitesimal functions as .

Find the limit of their ratio:

Since the limit equals 1, and are equivalent infinitesimal functions as .