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.
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6. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
7. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
8. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
9. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
lim
x→2
x2 − 4
x − 2
=
10. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
lim
x→2
x2 − 4
x − 2
= lim
x→2
11. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
lim
x→2
x2 − 4
x − 2
= lim
x→2
(x − 2)(x + 2)
x − 2
12. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
lim
x→2
x2 − 4
x − 2
= lim
x→2
$$$$(x − 2)(x + 2)
$$$x − 2
13. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
lim
x→2
x2 − 4
x − 2
= lim
x→2
$$$$(x − 2)(x + 2)
$$$x − 2
= lim
x→2
(x + 2)
14. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
lim
x→2
x2 − 4
x − 2
= lim
x→2
$$$$(x − 2)(x + 2)
$$$x − 2
= lim
x→2
(x + 2) = 2 + 2 =
15. Example 1
Let’s consider the limit:
lim
x→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
It equals 0
0!.
But we can factor:
lim
x→2
x2 − 4
x − 2
= lim
x→2
$$$$(x − 2)(x + 2)
$$$x − 2
= lim
x→2
(x + 2) = 2 + 2 = 4
34. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
35. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
36. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
lim
h→0
3a2h + 3ah2 + h3
h
37. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
lim
h→0
3a2h + 3ah2 + h3
h
= lim
h→0
38. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
lim
h→0
3a2h + 3ah2 + h3
h
= lim
h→0
h 3a2 + 3ah + h2
h
39. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
lim
h→0
3a2h + 3ah2 + h3
h
= lim
h→0
¡h 3a2 + 3ah + h2
¡h
40. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
lim
h→0
3a2h + 3ah2 + h3
h
= lim
h→0
¡h 3a2 + 3ah + h2
¡h
= lim
h→0
3a2
+ 3ah + h2
41. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
lim
h→0
3a2h + 3ah2 + h3
h
= lim
h→0
¡h 3a2 + 3ah + h2
¡h
= lim
h→0
3a2
+ 3ah + h2
= 3a2
+ 3a.0 + 02
42. Example 3
lim
h→0
(a + h)3 − a3
h
Here a is a constant. Let’s expand (a + h)3:
lim
h→0
a3 + 3a2h + 3ah2 + h3 − a3
h
=
lim
h→0
3a2h + 3ah2 + h3
h
= lim
h→0
¡h 3a2 + 3ah + h2
¡h
= lim
h→0
3a2
+ 3ah + h2
= 3a2
+ 3a.0 + 02
= 3a2