This is often given as a definition of e. See the Dr. Math reference, below, for a proof that the following three definitions of e are equivalent:
1. limn−>∞ (1+1/n)n = e
2. the unique real number greater than 1 such that the area under the curve y = 1/x from x = 1 to x = e is equal to 1.
3. e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...
let t = 1/(eh-1), so 1/t = eh-1, so 1+1/t = eh, so h = ln(1+1/t)
Now, limh−>0 (eh-1)/h
= limt−>∞ (1/t)/ln(1+1/t)
= limt−>∞ 1/(t ln(1+1/t))
= limt−>∞ 1/ln((1+1/t)t)
= 1/ln(e), which we know from the definition of e, above
If f(x) = ex, then f'(x) = limh−>0(ex+h-ex)/h
= limh−>0 ex(eh-1)/h
sinh(x)/x = (1/2)ex/x + (1/2)e-x/-x
limx−>0 sinh(x)/x = 1/2 + 1/2 = 1
Ask Dr. Math: http://mathforum.org/library/drmath/view/51954.html equivalence of the three commonly used definitions of e.
www.ies.co.jp/math/java/calc/exp/exp.html explanation of Limx−>∞ (1+1/x)x = e
www.ies.co.jp/math/java/calc/lim_e/lim_e.html explanation of Limx (ex-1)/x = 1
Proof that lim(x−>0) sin x / x = 1
Hyperbolic functions and the "Euler" section of the summary of trig equivalences
Derivative of Sin
Compound Interest - A = P(1+r/n)^(nt), continuous compounding A = Pe^(rt)
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