Honestly, e belongs in a calculus course, not in algebra 2. Presenting it prematurely makes it seem magical and incomprehensible. Why the rush to get e in before students are ready for it?
Lagrange's approach artfully downplays the limiting operation (indeed, it was his contention that this operation is not needed for the proper formulation of the calculus!) and does many marvelous things with "high school algebra" alone. Breathtaking brilliancies everywhere.
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The essential question, really, is: what polynomials come closest to satisfying the identity f(x)*f(y) = f(x + y)? One readily passes inductively from the nth-degree best approximation to the (n+1)st-degree best approximation. The answers all contain the same arbitrary constant.
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This constant may be chosen to be 1, which immediately simplifies everything marvelously, and feels like the obvious "best" value. Doing this corresponds to making e the base for the exponentiation operation. That is, it gives access to approximations e, by the values f(1).
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