Limits and nonstandard analysis - is my intuition correct?

Question

Having nonstandard analysis under our belts, would it be wrong to say that

$$\lim_{x\rightarrow a^{\pm ^{}}}f(x)$$

is the same thing as

$$f(x\pm ^{}{\mathrm{d} x})$$

where ${\mathrm{d} x}$ is an infinitesimal?

Answer 1

$\lim_{x \to a} f(x)$, if it exists, is the standard part of $f(a+dx)$ for every nonzero infinitesimal $dx$. Similarly, $\lim_{x \to a^+} f(x)$, if it exists, is the standard part of $f(a+dx)$ for every positive infinitesimal $dx$. Note the quantifier: the limit only exists if the standard part is the same no matter what infinitesimal you choose.

We don't really use limits very much in the hyperreals.