### 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^{x}e^{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)

The webmaster and author of this Math Help site is
Graeme McRae.