the Number e as a Limit

ex =lim

n→ ∞1 + x

n⎛

⎝⎜

⎞

⎠⎟

n

The Number e as a Limit by M. Seppälä

ex

The mathematical constant e is that

number for which the tangent to the graph of ex at x = 0 has

the slope 1.

e ≈ 2.718281828

Exponentials

The Number e as a Limit by M. Seppälä

Euler’s Argument

N =x εFor any given positive number x, is infinitely large.

For infinitely small ε, eε =1 + ε.

ex =eNε

= eε

( )N

= 1 + ε( )

N

ex =lim

n→ ∞1 +

xn

⎛

⎝⎜⎞

⎠⎟

n = 1 +

xN

⎛

⎝⎜⎞

⎠⎟

N

The Number e as a Limit by M. Seppälä

The Number e as a LimitA consequence of the definition of the mathematical constant e was that D(ex) = ex.

By the Inverse Function Rule, this implies that D(ln x) = 1/x.

In particular,

d ln x( )

dxx=1

=1.

The Number e as a Limit by M. Seppälä

The Number e as a Limit

implies

limh→ 0

ln 1 + h( ) −ln1

h=1 ⇔ lim

h→ 0ln 1 + h( )

1h

⎛

⎝⎜

⎞

⎠⎟ =1.

The formula

d ln x( )

dxx=1

=1

The Number e as a Limit by M. Seppälä

limh→ 0

ln 1 + h( )1h

⎛

⎝⎜

⎞

⎠⎟ =ln lim

h→ 01 + h( )

1h

⎛

⎝⎜

⎞

⎠⎟ =1.

lim

h→ 01 + h( )

1h =e.

Since ln is a continuous function,

This implies

We have shown that .

THE NUMBER E AS A LIMIT

The Number e as a Limit by M. Seppälä

lim

h→ 01 + h( )

1h =e.

Equivalently:

We have shown that

THE EXPONENTIAL FUNCTION

lim

n→ ∞1 +

1n

⎛

⎝⎜⎞

⎠⎟

n

=e.

The Number e as a Limit by M. Seppälä

1 +

xn

⎛

⎝⎜⎞

⎠⎟

n

Hence

THE EXPONENTIAL FUNCTION

= 1 +1nx

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

nx

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

x

n→ ∞⏐ →⏐ ⏐ ex .

ex =lim

n→ ∞1 +

xn

⎛

⎝⎜⎞

⎠⎟

n

LEONHARD EULER (1707 - 1783)