Limits by Factoring
pabloantuna75
495 views
43 slides
May 17, 2013
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About This Presentation
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.html
Slide Content
Example 1
Example 1
Let's consider the limit:
Let's consider the limit:
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
Let's consider the limit:
lim
x!2
x
2
4
x2
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
=
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
=
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
= lim
x!2
(x+ 2)
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
= lim
x!2
(x+ 2)
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
= lim
x!2
(x+ 2) = 2 + 2 =
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
= lim
x!2
(x+ 2) = 2 + 2 =
Example 1
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
= lim
x!2
(x+ 2) = 2 + 2 =
Let's consider the limit:
lim
x!2
x
2
4
x2
We note that atx= 2, our function is indeterminate.
It equals
0
0
!.
But we can factor:
lim
x!2
x
2
4
x2
= lim
x!2
(x2)(x+ 2)
x2
= lim
x!2
(x+ 2) = 2 + 2 =
Example 1
Example 1
What is the graph of this function?
What is the graph of this function?
Example 1
What is the graph of this function?
f(x) =
x
2
2
x+ 2
What is the graph of this function?
f(x) =
x
2
2
x+ 2
Example 1
What is the graph of this function?
f(x) =
x
2
2
x+ 2
What is the graph of this function?
f(x) =
x
2
2
x+ 2
Example 1
What is the graph of this function?
f(x) =
x
2
2
x+ 2
It is the graph ofx+ 2, but with a!
What is the graph of this function?
f(x) =
x
2
2
x+ 2
It is the graph ofx+ 2, but with a!
Example 2
Example 2
lim
x!1
x
3
1
x1
lim
x!1
x
3
1
x1
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
=
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
=
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
= lim
x!1
x
2
+x+ 1
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
= lim
x!1
x
2
+x+ 1
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
= lim
x!1
x
2
+x+ 1
= 1
2
+ 1 + 1
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
= lim
x!1
x
2
+x+ 1
= 1
2
+ 1 + 1
Example 2
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
= lim
x!1
x
2
+x+ 1
= 1
2
+ 1 + 1 =
lim
x!1
x
3
1
x1
Remember how to factor the numerator?
lim
x!1
x
3
1
x1
= lim
x!1
(x1)(x
2
+x+ 1)
x1
= lim
x!1
x
2
+x+ 1
= 1
2
+ 1 + 1 =
Example 3
Example 3
lim
h!0
(a+h)
3
a
3
h
lim
h!0
(a+h)
3
a
3
h
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
= lim
h!0
3a
2
+ 3ah+h
2
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
= lim
h!0
3a
2
+ 3ah+h
2
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
= lim
h!0
3a
2
+ 3ah+h
2
= 3a
2
+ 3a:0 + 0
2
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
= lim
h!0
3a
2
+ 3ah+h
2
= 3a
2
+ 3a:0 + 0
2
Example 3
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
= lim
h!0
3a
2
+ 3ah+h
2
= 3a
2
+ 3a:0 + 0
2
=a
2
lim
h!0
(a+h)
3
a
3
h
Hereais a constant. Let's expand (a+h)
3
:
lim
h!0
a
3
+ 3a
2
h+ 3ah
2
+h
3
a
3
h
=
lim
h!0
3a
2
h+ 3ah
2
+h
3
h
= lim
h!0
h
3a
2
+ 3ah+h
2
h
= lim
h!0
3a
2
+ 3ah+h
2
= 3a
2
+ 3a:0 + 0
2
=a
2
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