limits involving number e

63

Upload: pablo-antuna

Post on 30-Jun-2015

134 views

Category:

Education


2 download

DESCRIPTION

In this presentation we learn to solve limits using the limit definition of number e. For more lessons and videos: http://www.intuitive-calculus.com/solving-limits.html

TRANSCRIPT

Page 1: Limits Involving Number e
Page 2: Limits Involving Number e
Page 3: Limits Involving Number e

The Number e

Page 4: Limits Involving Number e

The Number e

The number e is defined as:

Page 5: Limits Involving Number e

The Number e

The number e is defined as:

limn→∞

(1 +

1

n

)n

Page 6: Limits Involving Number e

The Number e

The number e is defined as:

limn→∞

(1 +

1

n

)n

This number appears in many branches of science andmathematics.

Page 7: Limits Involving Number e

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.

Page 8: Limits Involving Number e

Example 1

Page 9: Limits Involving Number e

Example 1

Let’s say we want to find this limit:

Page 10: Limits Involving Number e

Example 1

Let’s say we want to find this limit:

limn→∞

(1 +

1

n

)n+5

Page 11: Limits Involving Number 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:

Page 12: Limits Involving Number 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→∞

Page 13: Limits Involving Number 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

Page 14: Limits Involving Number 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

Page 15: Limits Involving Number 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

= limn→∞

(1 +

1

n

)n

. limn→∞

(1 +

1

n

)5

Page 16: Limits Involving Number 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

=��

������:

elimn→∞

(1 +

1

n

)n

. limn→∞

(1 +

1

n

)5

Page 17: Limits Involving Number 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

. limn→∞

1 +����0

1

n

5

Page 18: Limits Involving Number 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

Page 19: Limits Involving Number 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

Page 20: Limits Involving Number 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

Page 21: Limits Involving Number 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

Page 22: Limits Involving Number e

Example 2

Page 23: Limits Involving Number e

Example 2

limx→∞

(1 +

1

x

)4x

Page 24: Limits Involving Number e

Example 2

limx→∞

(1 +

1

x

)4x

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

Page 25: Limits Involving Number e

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

Page 26: Limits Involving Number e

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→∞

Page 27: Limits Involving Number e

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

.

Page 28: Limits Involving Number e

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

.

Page 29: Limits Involving Number e

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

.

Page 30: Limits Involving Number e

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

Page 31: Limits Involving Number e

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

Page 32: Limits Involving Number e

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

Page 33: Limits Involving Number e

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

Page 34: Limits Involving Number e

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

Page 35: Limits Involving Number e

Example 3

Page 36: Limits Involving Number e

Example 3

limx→∞

(x + 3

x − 1

)x+3

Page 37: Limits Involving Number e

Example 3

limx→∞

(x + 3

x − 1

)x+3

This looks somewhat like the definition of e.

Page 38: Limits Involving Number 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!

Page 39: Limits Involving Number 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!

1 +1

y=

x + 3

x − 1

Page 40: Limits Involving Number 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!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

Page 41: Limits Involving Number 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!

1 +1

y=

x + 3

x − 1

Here we need to solve for x :

1

y=

x + 3

x − 1− 1

Page 42: Limits Involving Number 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!

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

Page 43: Limits Involving Number 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!

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

Page 44: Limits Involving Number 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!

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

Page 45: Limits Involving Number 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!

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

Page 46: Limits Involving Number 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!

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

Page 47: Limits Involving Number 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!

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

Page 48: Limits Involving Number e

Example 3

Page 49: Limits Involving Number e

Example 3

Let’s now replace in our limit:

Page 50: Limits Involving Number e

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

Page 51: Limits Involving Number e

Example 3

Let’s now replace in our limit:

limx→∞

(x + 3

x − 1

)x+3

1 + 1y = x+3

x−1

Page 52: Limits Involving Number e

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

Page 53: Limits Involving Number e

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 →∞?

Page 54: Limits Involving Number e

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!

Page 55: Limits Involving Number e

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

Page 56: Limits Involving Number e

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→∞

Page 57: Limits Involving Number e

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

Page 58: Limits Involving Number e

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

Page 59: Limits Involving Number e

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

Page 60: Limits Involving Number e

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

Page 61: Limits Involving Number e

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

Page 62: Limits Involving Number e

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

Page 63: Limits Involving Number e