Suppose you know that the position of an object depends on time as
(here I'm being naughty and forgetting about units for
the moment). Lets calculate the instantaneous velocity at t = 1.
So in this case 
, and 
. We'll want to try
different values of 
and verify that we do appear to converge
to a sensible final answer.
Let's start with 
, then 
so
from eq. 3.1 we have,
over this time interval is
OK that's fine, but this is clearly not an infinitesimal interval.
Let's shrink the interval by 1/2 so that 
. Then
If we shrink the interval 
even further, so that
then going through the same steps gives
. If we now try 
, then 
.
It looks pretty clear that as 
we're coming
up with an instantaneous velocity of 2.
This is what you'd expect since the derivative of 
is 2t .
Evaluating this at t=1, we get 2.