Definition of e (Mathematical Constant)

Definition of e (Mathematical Constant)

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

Definition

The limit of (1 + x)^(1/x) as x goes to 0 is e. The limit of (1 + 1/x)^x as x goes to infinity is also e.

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

Find the limit of the given expression. The limit of (1 + 7x)^(1/x) as x goes to 0.

The base is 1 + 7x.

So, change the exponent 1/x into 1/7x.
(the reciprocal of 7x)

And, to undo the 7,
multiply 7 to the exponent.

Then, as x → 0,
the blue part becomes e.

So (limit) = e7.

Example 2

Find the limit of the given expression. The limit of (1 + 5/x)^x as x goes to infinity.

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) = e5.

Example 3

Find the limit of the given expression. The limit of [ln (1 + 2x)]/x as x goes to 0.

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 + 2x.

So, change the exponent 1/x into 1/2x.
(the reciprocal of 2x)

And, to undo the 2,
multiply 2 to the exponent.

Then, as x → 0,
the blue part becomes e.

So (limit) = ln (e2).

Take the exponent 2 out.

Logarithms of powers

So 2 ln e = 2.

Natural logarithms