# Constant e

How to use the definition of the constant e to solve the given limit: definition, 2 examples, and their solutions.

## Definition

The limit of (1 + x)^{1/x} as x → 0 is

the constant e.

Just like π = 3.141592...,

e is a special constant:

e = 2.71818... .

e is also called

the Euler's number, natural base, natural constant.

The limit of (1 + 1/x)^{x} as x → ∞ is

also the constant e.

## Examplelim_{x → 0} (1 + 7x)^{1/x}

First write the limit part and (1 + 7x).

The x term of (1 + 7x) is 7x.

So write, the reciprocal of 7x, 1/7x

in the exponent.

The exponent of the given expression is 1/x.

But you wrote 1/7x.

So, to undo the denominator 7,

multiply 7.

So (1 + 7x)^{1/x} = (1 + 7x)^{[1/7x]⋅7}.

As x → 0,

(1 + 7x)^{1/7x} → e

and write the exponent 7.

So e^{7} is the answer.

## Examplelim_{x → ∞} (1 + 5/x)^{x}

First write the limit part and (1 + 5/x).

The 1/x term of (1 + 5/x) is 5/x.

So write, the reciprocal of 5/x, x/5

in the exponent.

The exponent of the given expression is x.

But you wrote x/5.

So, to undo the denominator 5,

multiply 5.

So (1 + 5/x)^{x} = (1 + 5/x)^{[x/5]⋅5}.

As x → ∞,

(1 + 5/x)^{x/5} → e

and write the exponent 5.

So e^{5} is the answer.