I was teaching a friend a bit of Calculus and we came across this problem:
Given the definition $\epsilon, \delta$ of derivatives, we were searching the derivative of the function $x^2+x$ at the point $x_0=1$. So we expanded in terms of $\Delta x$ and we got $\lim_{\Delta x\to0} (\Delta x + 3)=3$, solving the problem. But he asked me why $\Delta x$ goes to zero, in the sense of disappearing in the RHS. I said to him I thought that's because of the definition of limit and I said a ''proof'' for $\lim_{\Delta x\to0} \Delta x=0$ would be something like this: for all $\epsilon>0$ there exists $\delta >0$ such that $0<|\Delta x-0|<\delta$ implies $0<|\Delta x-0|<\epsilon$, which is true by taking $\delta=\epsilon$, and therefore the limit is zero. I think this is similar to the increment $h$ used in some textbooks I've read. But our doubt was (is): isn't $\Delta x$ a function of $x$ and if so could we use it to prove the limit as I did? I've never noticed such case, and couldn't find anywhere on the web... So I appreciate any help! Thank you!