# Is $\lim_{x\to 0} (x)$ different from $dx$

I think the title pretty much explains what I want to ask. Bassically is $$\lim_{x\to 0}(x)$$ different from $dx$?

Anoher way to put this would be, how wouldth equation: $$F=adm$$ Be different from $$F=a\lim_{m\to 0}(m)$$ I think the second can be written as $$F=0$$ Whilst the first we cannot, but I cannot explain why?

• Whatever else $dx$ is, it is not a real number. The limit of $x$ as $x \to 0$ is a real number. These two quantities cannot be equal. Feb 20, 2015 at 19:28
• @SimonS sorry why is $dx$ not a real number? Or equivlently If I have an infitismal bit of mass $dm$ why is this not a real number? Feb 20, 2015 at 19:29
• I was hoping that the title didn't explain well what you wanted to ask.
– user207710
Feb 20, 2015 at 19:29
• Joseph, how do you define an infinitesimal? How do you define arithmetic operations on those? Now, do these definitions fit with the usual real numbers definitions? This should answer your question. Feb 20, 2015 at 19:32
• @bartgol I get your (and Simon S's) point but I can't see when we are using it to define real quantites, such as an infitimsal bit of mass, or an infitismal length of string, how it cannot be a real number? Feb 20, 2015 at 19:34

It is confusing because the way derivatives are taught today are different from how it was done back in the 1600s. Back then a derivative was $dy/dx$, where $dy$ and $dx$ were infinitesimal quantities. Today, we introduce derivatives using limits. The confusing part is that learn differential calculus by limits but use the old notation.