Epsilon-Delta Limit Proof
what are the rest of your decimals? unlkess you tell us exactly what number e is, you cannot prove that e^x really approaches e as x goes to 1 by your method. and unless you define precisely what e^x means you also cannot do it.
the easiest way is halls second method that e^x is continuous, since it is the inverse of the continuous function ln(x), hence it suffices to show e^1 = e, but that follows from the fact that e is defined as the unique number such that ln(e) = 1.