limits by factoring
DESCRIPTION
In this video we learn how to solve limits by factoring and cancelling. This is one of the most simple and powerful techniques for solving limits. Watch video: http://www.youtube.com/watch?v=r0Qw5gZuTYE For more videos and lessons: http://www.intuitive-calculus.com/solving-limits.htmlTRANSCRIPT
![Page 1: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/1.jpg)
![Page 2: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/2.jpg)
![Page 3: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/3.jpg)
Example 1
![Page 4: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/4.jpg)
Example 1
Let’s consider the limit:
![Page 5: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/5.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
![Page 6: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/6.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
![Page 7: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/7.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.
![Page 8: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/8.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
![Page 9: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/9.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
limx→2
x2 − 4
x − 2=
![Page 10: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/10.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
limx→2
x2 − 4
x − 2= lim
x→2
![Page 11: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/11.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
limx→2
x2 − 4
x − 2= lim
x→2
(x − 2)(x + 2)
x − 2
![Page 12: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/12.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
limx→2
x2 − 4
x − 2= lim
x→2
����(x − 2)(x + 2)
���x − 2
![Page 13: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/13.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
limx→2
x2 − 4
x − 2= lim
x→2
����(x − 2)(x + 2)
���x − 2
= limx→2
(x + 2)
![Page 14: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/14.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
limx→2
x2 − 4
x − 2= lim
x→2
����(x − 2)(x + 2)
���x − 2
= limx→2
(x + 2) = 2 + 2 =
![Page 15: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/15.jpg)
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.It equals 0
0 !.But we can factor:
limx→2
x2 − 4
x − 2= lim
x→2
����(x − 2)(x + 2)
���x − 2
= limx→2
(x + 2) = 2 + 2 = 4
![Page 16: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/16.jpg)
Example 1
![Page 17: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/17.jpg)
Example 1
What is the graph of this function?
![Page 18: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/18.jpg)
Example 1
What is the graph of this function?
f (x) =x2 − 2
x + 2
![Page 19: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/19.jpg)
Example 1
What is the graph of this function?
f (x) =x2 − 2
x + 2
![Page 20: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/20.jpg)
Example 1
What is the graph of this function?
f (x) =x2 − 2
x + 2
It is the graph of x + 2, but with a hole!
![Page 21: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/21.jpg)
Example 2
![Page 22: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/22.jpg)
Example 2
limx→1
x3 − 1
x − 1
![Page 23: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/23.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
![Page 24: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/24.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1=
![Page 25: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/25.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1= lim
x→1
![Page 26: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/26.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1= lim
x→1
(x − 1)(x2 + x + 1)
x − 1
![Page 27: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/27.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1= lim
x→1
����(x − 1)(x2 + x + 1)
���x − 1
![Page 28: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/28.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1= lim
x→1
����(x − 1)(x2 + x + 1)
���x − 1
= limx→1
(x2 + x + 1
)
![Page 29: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/29.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1= lim
x→1
����(x − 1)(x2 + x + 1)
���x − 1
= limx→1
(x2 + x + 1
)= 12 + 1 + 1
![Page 30: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/30.jpg)
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1= lim
x→1
����(x − 1)(x2 + x + 1)
���x − 1
= limx→1
(x2 + x + 1
)= 12 + 1 + 1 = 3
![Page 31: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/31.jpg)
Example 3
![Page 32: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/32.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
![Page 33: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/33.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
![Page 34: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/34.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
a3 + 3a2h + 3ah2 + h3 − a3
h
![Page 35: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/35.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
![Page 36: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/36.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
limh→0
3a2h + 3ah2 + h3
h
![Page 37: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/37.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
limh→0
3a2h + 3ah2 + h3
h= lim
h→0
![Page 38: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/38.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
limh→0
3a2h + 3ah2 + h3
h= lim
h→0
h(3a2 + 3ah + h2
)h
![Page 39: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/39.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
limh→0
3a2h + 3ah2 + h3
h= lim
h→0
�h(3a2 + 3ah + h2
)�h
![Page 40: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/40.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
limh→0
3a2h + 3ah2 + h3
h= lim
h→0
�h(3a2 + 3ah + h2
)�h
= limh→0
(3a2 + 3ah + h2
)
![Page 41: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/41.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
limh→0
3a2h + 3ah2 + h3
h= lim
h→0
�h(3a2 + 3ah + h2
)�h
= limh→0
(3a2 + 3ah + h2
)= 3a2 + 3a.0 + 02
![Page 42: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/42.jpg)
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
limh→0
��a3 + 3a2h + 3ah2 + h3 −��a3
h=
limh→0
3a2h + 3ah2 + h3
h= lim
h→0
�h(3a2 + 3ah + h2
)�h
= limh→0
(3a2 + 3ah + h2
)= 3a2 + 3a.0 + 02 = 3a2
![Page 43: Limits by Factoring](https://reader033.vdocuments.mx/reader033/viewer/2022042614/559288521a28ab4e468b4755/html5/thumbnails/43.jpg)