Idea of a Limit When I first learned Calculus, I was always taught the infinitesimal concept of a limit in regards to slope. It was not until awhile later that I came across the epsilon-delta, $(\epsilon, \delta)$, definition of a limit. From a number theorist's standpoint, I am wanting to know if the infinitesimal definition should be even taught to students taking Calculus I. Obviously there are patterns between the two, but I was wanting to know if the idea of an infinitesimal has any merit to it in regards to proofs. All opinions are welcome in regard to this question; I am just wanting to see other people's ideas.
 A: If you read about the details of the history of calculus from Galileo to Newton to Euler, especially about the  criticisms of it, you will see why the infinitesimals were exterminated in the 19th century, to put the topic on a rigorous logical foundation. There had not even been an axiomatic definition of $\mathbb R.$  In the 20th century the infinitesimals were re-invented by extending $\mathbb R$ to larger ordered fields.
Classical results can be re-interpreted. For example, when $d\ne 0$, for brevity  let $D(x,d)= \frac {(x+d)^2-x^2}{(x+d)-x}=2x+d.$ In an ordered-field extension of $\mathbb R $ with positive members that are less than any positive member of $\mathbb R$ we can say literally:  
When $d\ne 0$ and $|d|$ is infinitely small then $D(x,d)$ is infinitely close to $2x.$
Here, "$|d|$ is infinitely small" means $|d|<r$ for all $r\in \mathbb R^+,$ and "$D(x,d)$ is infinitely close to $2x$" means $|D(x,d)-2x|<r$ for all $r\in \mathbb R^+.$     
A nice property of Liebnitz' notation $dy/dx$ and $\int f(x)dx ,$ etc. is that in many cases we can treat $dx,$ etc. as if it were a number, and can justify it within a classical context, or, in a different manner, in the context of an extension of $\mathbb R.$ But like most free offers, some restrictions apply.
