Going way back to definitions of e, my favorite is lim(x->∞) (1 + 1/x)^x = e (I have no idea how you are supposed to write limits in horizontal form )
I write them the same way you did, since it is more legible than TeX.
As a minor nitpick, I would say e = lim(n->∞)(1+1/n)^n. The reason is because the variable n is implicitly assumed to be a whole number, while x is implicitly assumed to be an arbitrary real number. While both statements are true, defining irrational powers is a bit subtle. In fact, one common approach is to first define exp(x) and ln(x) without reference to irrational exponents, and then define a^x = exp(x ln(a)).