TITLE: Is the following correct $dx = \lim_{ \Delta x\to 0} \Delta x\ $?
QUESTION [6 upvotes]: I hope your answer for this question will help me understand a lot of things. Thank you guys.

REPLY [6 votes]: Under conventional definitions $\lim\limits_{\Delta x\,\to\,0} \Delta x =0,$ whereas $dx,$ is not really defined in a logically rigorous way under currently conventional definitions, but it has a traditional (as opposed to either conventional or logically rigorous) definition. The quantity $dx$ was introduced int he 1600s by Leibniz and is supposed to represent an infinitely small change in $x$. If $y$ is a function of $x$ then $dy$ is the corresponding infinitely small change in $y$.
Even though $\lim\limits_{\Delta x\,\to\,0} \Delta x=0$ and consequently $\lim\limits_{\Delta y\,\to\,0} \Delta y =0,$ nonetheless $\lim\limits_{\Delta x\,\to\,0} \dfrac{\Delta y}{\Delta x}$ is some particular ordindary, usually nonzero, finite number.