# Definition of *e* (Mathematical Constant)

How to solve the limit problems by using the definition of e: definition, examples, and their solutions.

## Definition

*e* is defined as (1 + 0)^{∞}.

So the limit of (1 + *x*)^{(1/x)} as *x* → 0 is *e*.

And the limit of (1 + 1/*x*)^{x} as *x* → ∞ is also *e*.*e* = 2.718... is a mathematical constant

that is widely used in various fields:

mathematics, science, engineering, finance, etc.

## Example 1

The base is 1 + 7*x*.

So, change the exponent 1/*x* into 1/7*x*.

(the reciprocal of 7*x*)

And, to undo the 7,

multiply 7 to the exponent.

Then, as *x* → 0,

the blue part becomes *e*.

So (limit) = *e*^{7}.

## Example 2

The base is 1 + 5/*x*.

So, change the exponent *x* into *x*/5.

(the reciprocal of 5/*x*)

And, to undo the '/5',

multiply 5 to the exponent.

Then, as *x* → ∞,

the blue part becomes *e*.

So (limit) = *e*^{5}.

## Example 3

Move the denominator *x*

to the front of the natural log.

Then move 1/*x* into the base's exponent.

Logarithms of powers

The base is 1 + 2*x*.

So, change the exponent 1/*x* into 1/2*x*.

(the reciprocal of 2*x*)

And, to undo the 2,

multiply 2 to the exponent.

Then, as *x* → 0,

the blue part becomes *e*.

So (limit) = ln (*e*^{2}).

Take the exponent 2 out.

Logarithms of powers

So 2 ln *e* = 2.

Natural logarithms