Constant e

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

Definition

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.

Example 1

Example

Solution

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 e7 is the answer.

Example 2

Example

Solution

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 e5 is the answer.