An infinitesimal is so small that no matter how small you think it is, it is always smaller than what you thought.
An infinitesimal is approaching zero. It won’t get there but it brags that it is closer than you.
Let x+2=5.
Let x be less than 3 and let x be approaching 3.
Let dx=3-x.
If x=2.999 then dx=0.001. You can probably see that if we add more lines to ask we will get more zeros in dx between the decimal and the one.
We can make dx as small as we want by moving x closer and closer to 3.
Isaac Newton comes along and says:
If, while x is on an interval close to 3 and approaching 3, we can get (x+2) as close to 5 as we want, by moving x closer to 3, then when x=3, (x+2)=5.
In a strange way, the idea depends on us NOT trying to define the adjective ‘close’. Close is close if it is close enough.
The above is a simplified example of a core idea did Isaac Newton put forth. Give up is not quite Isaac’s idea. In his example he talked about ratios rather than values. He would have dy/dx approaching something. Also, instead of saying that x was approaching 3, he would say that 3 – x was approaching zero.
Obviously we know that 2+3=5. In problems where this idea is likely to be used, we will have a situation where we can’t set X to be equal to the goal because at that point we get a division by zero error.
Cauchy Definition of a limit
When the successively attributed values of the same variable indefinitely approach a fixed value, so that
finally they differ from it by as little as desired, the last is called the limit of all the others.
Vectoreverything 