Define:
    e = lim[ (1+ h)^1/h ] as h 0.

Then let:
    (1+ h)^1/h / e = 1 + Δ where Δ 0 as h 0.

Then
    (1 + h) = (1 + Δ)^h e^h.
hence
    [ (e^h - 1) / h ] = [ (1 + h) / (1 + Δ)^h - 1 ] / h = [ (1 + h) / (1 + Δ)^-h - 1 ] / h

The binomial theorem gives:
    (1 + Δ)^-h = 1 - h Δ + higher order terms
so that
    [ (e^h - 1) / h ] = [ (1 + h) (1 - h Δ + ...) - 1 ] / h = 1 - Δ + ...

Hence
    [ (e^h - 1) / h ] 1 as h, hence Δ, approaches 0.

Finally:
    d/dx(e^x) = lim[ (e^x+h - e^x) / h ] = e^x lim[ (e^h - 1) / h ] = e^x as h 0.

See: Math Stuff