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 Skip Navigation LinksMath Help > Calculus > Limit > Exponential Limit

Limn−>∞ (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. 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! + ...

Limh (eh-1)/h = 1

Proof:

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
   = 1

Application to the derivative of ex

If f(x) = ex, then f'(x) = limh−>0(ex+h-ex)/h
   = limh−>0(exeh-ex)/h
   = limh−>0 ex(eh-1)/h
   = ex 

Application to the limx−>0 sinh(x)/x

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

limx−>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 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

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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