Limits Involving Number e

The Number e

The Number e

The number e is defined as:

The Number e

The number e is defined as:

limn→∞

(1 +

1

n

)n

The Number e

The number e is defined as:

limn→∞

(1 +

1

n

)n

This number appears in many branches of science andmathematics.

The Number e

The number e is defined as:

limn→∞

(1 +

1

n

)n

This number appears in many branches of science andmathematics.Here we’ll use its definition to calculate other limits.

Example 1

Example 1

Let’s say we want to find this limit:

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

= limn→∞

(1 +

1

n

)n

. limn→∞

(1 +

1

n

)5

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

=��

������:

elimn→∞

(1 +

1

n

)n

. limn→∞

(1 +

1

n

)5

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

=���

�����:e

limn→∞

(1 +

1

n

)n

. limn→∞

1 +����0

1

n

5

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

=���

�����:e

limn→∞

(1 +

1

n

)n

.��

������:

1limn→∞

(1 +

1

n

)5

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

=���

�����:e

limn→∞

(1 +

1

n

)n

.��

������:

1limn→∞

(1 +

1

n

)5

= e.1

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

=���

�����:e

limn→∞

(1 +

1

n

)n

.��

������:

1limn→∞

(1 +

1

n

)5

= e.1 = e

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

We use the properties of exponents and that of limits of products:

= limn→∞

(1 +

1

n

)n

.

(1 +

1

n

)5

=���

�����:e

limn→∞

(1 +

1

n

)n

.��

������:

1limn→∞

(1 +

1

n

)5

= e.1 = e

Example 2

Example 2

limx→∞

(1 +

1

x

)4x

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

(1 +

1

x

)x

.

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

.

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

=

[limx→∞

(1 +

1

x

)x]4

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

=

[���

�����:e

limx→∞

(1 +

1

x

)x]4

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

=

[���

�����:e

limx→∞

(1 +

1

x

)x]4= e4

Example 2

limx→∞

(1 +

1

x

)4x

Here we use again the properties of exponents and those of limits:

= limx→∞

[(1 +

1

x

)x]4= lim

x→∞

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

.

(1 +

1

x

)x

=

[���

�����:e

limx→∞

(1 +

1

x

)x]4= e4

Example 3

Example 3

limx→∞

(x + 3

x − 1

)x+3

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1 =

x + 3− x + 1

x − 1

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1 =

�x + 3−�x + 1

x − 1

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1 =

�x + 3−�x + 1

x − 1=

4

x − 1

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1 =

�x + 3−�x + 1

x − 1=

4

x − 1

x − 1 = 4y

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1 =

�x + 3−�x + 1

x − 1=

4

x − 1

x − 1 = 4y ⇒ x = 4y + 1

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.So, let’s make a change of variables!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1 =

�x + 3−�x + 1

x − 1=

4

x − 1

x − 1 = 4y ⇒ x = 4y + 1

Example 3

Example 3

Let’s now replace in our limit:

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

= limy→∞

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

= limy→∞

(1 +

1

y

)4y+4

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

= limy→∞

(1 +

1

y

)4y+4

=

[limy→∞

(1 +

1

y

)y]4. limy→∞

(1 +

1

y

)4

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

= limy→∞

(1 +

1

y

)4y+4

=

[���

�����:e

limy→∞

(1 +

1

y

)y]4. limy→∞

(1 +

1

y

)4

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

= limy→∞

(1 +

1

y

)4y+4

=

[���

�����:e

limy→∞

(1 +

1

y

)y]4.���

�����:1

limy→∞

(1 +

1

y

)4

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

= limy→∞

(1 +

1

y

)4y+4

=

[���

�����:e

limy→∞

(1 +

1

y

)y]4.���

�����:1

limy→∞

(1 +

1

y

)4

= e4

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1 x = 4y + 1

What happens with y when x →∞?It will also approach infinity!

limx→∞

(x + 3

x − 1

)x+3

= limy→∞

(1 +

1

y

)4y+4

=

[���

�����:e

limy→∞

(1 +

1

y

)y]4.���

�����:1

limy→∞

(1 +

1

y

)4

= e4