Upload
pablo-antuna
View
136
Download
2
Embed Size (px)
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
Citation preview
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