Lim_n−>∞ (1+1/n)ⁿ = e

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. lim_n−>∞ (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! + ...

Lim_h (e^h-1)/h = 1

Proof:

let t = 1/(e^h-1), so 1/t = e^h-1, so 1+1/t = e^h, so h = ln(1+1/t)

Now, lim_h−>0 (e^h-1)/h
   = lim_t−>∞ (1/t)/ln(1+1/t)
   =  lim_t−>∞ 1/(t ln(1+1/t))
   = lim_t−>∞ 1/ln((1+1/t)^t)
   = 1/ln(e), which we know from the definition of e, above
   = 1

Application to the derivative of e^x

If f(x) = e^x, then f'(x) = lim_h−>0(e^x+h-e^x)/h
   = lim_h−>0(e^xe^h-e^x)/h
   = lim_h−>0 e^x(e^h-1)/h
   = e^x 

Application to the lim_x−>0 sinh(x)/x

sinh(x)/x = (1/2)e^x/x + (1/2)e^-x/-x

lim_x−>0 sinh(x)/x = 1/2 + 1/2 = 1

Internet references

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 Lim_x−>∞ (1+1/x)^x = e

www.ies.co.jp/math/java/calc/lim_e/lim_e.html explanation of Lim_x (e^x-1)/x = 1

Related pages in this website

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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