Download - Limits Involving Number e
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The Number e
![Page 4: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/4.jpg)
The Number e
The number e is defined as:
![Page 5: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/5.jpg)
The Number e
The number e is defined as:
limn→∞
(1 +
1
n
)n
![Page 6: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/6.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/7.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/8.jpg)
Example 1
![Page 9: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/9.jpg)
Example 1
Let’s say we want to find this limit:
![Page 10: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/10.jpg)
Example 1
Let’s say we want to find this limit:
limn→∞
(1 +
1
n
)n+5
![Page 11: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/11.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/12.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/13.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/14.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/15.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/16.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/17.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/18.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/19.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/20.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/21.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/22.jpg)
Example 2
![Page 23: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/23.jpg)
Example 2
limx→∞
(1 +
1
x
)4x
![Page 24: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/24.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/25.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/26.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/27.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/28.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/29.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/30.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/31.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/32.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/33.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/34.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/35.jpg)
Example 3
![Page 36: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/36.jpg)
Example 3
limx→∞
(x + 3
x − 1
)x+3
![Page 37: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/37.jpg)
Example 3
limx→∞
(x + 3
x − 1
)x+3
This looks somewhat like the definition of e.
![Page 38: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/38.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/39.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/40.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/41.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/42.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/43.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/44.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/45.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/46.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/47.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/48.jpg)
Example 3
![Page 49: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/49.jpg)
Example 3
Let’s now replace in our limit:
![Page 50: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/50.jpg)
Example 3
Let’s now replace in our limit:
limx→∞
(x + 3
x − 1
)x+3
![Page 51: Limits Involving Number e](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/51.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/52.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/53.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/54.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/55.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/56.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/57.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/58.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/59.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/60.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/61.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/62.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022060120/5592887e1a28ab48468b4792/html5/thumbnails/63.jpg)