 Consider the limit of the quantity \(((x+h)^3-x^3)/h\) as \(h\) approaches 0. In the standard reals, to show that this is \(3x^2\), we need to show that for every \(\epsilon\) there is a \(\delta\) such that for any value of \(h\) less than \(\delta\), the corresponding value of this function is less than \(\epsilon\) away from \(3x^2\). In this case, it turns out that choosing \(\delta &lt; \epsilon/4x\) works when \(x\) is sufficiently large, and choosing \(\delta &lt; \epsilon\) works when \(x\) is sufficiently small, but figuring out these choices is difficult.