A way of understanding the $dx$

Question

In an attempt to explain the concept of the infinitesimal change, I have defined it as such :
Given an interval of size $L$, we could express $L$ as follows :$L=\alpha.dx$ thus, theoretically, $\alpha = \frac{L}{dx}$ could be seen as the total number of $x$s since $dx$ is infinitesimally small.
Do you think this to be a conceptually wrong explanation?

Answer 1

Let consider the interval of length $L$ divided in $n$ equal parts of length $\Delta x$ therefore we have

$$\Delta x= \frac L n \iff L=n\cdot \Delta x$$

we can then define the infinitesimal interval as

$$dx:=\lim_{n\to \infty }\Delta x= \lim_{n\to \infty }\frac L n= \frac L {\lim_{n\to \infty }n}=0$$

but it is meaningless write $$dx=\frac L \infty$$

since $\infty$ is not a number

Answer 2

In concrete terms, an infinitesimal change is so small that the linear approximation holds.

The local behavior of a smooth function can be described as

$$f(x+h)=f(x)+ah+r(x,h)$$

where $a$ is a constant and $r$ a remainder term that vanishes when $h$ tends to zero, $\lim_{h\to0}\dfrac{r(x,h)}h=0$.

Then $h$ infinitesimal means that we can just drop the remainder term and "admit" the equality

$$f(x+h)=f(x)+ah.$$