Example 1
Example 1
Let’s consider the limit:
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
Example 1
Let’s consider the limit:
limx→2
x2 − 4
x − 2
We note that at x = 2, our function is indeterminate.
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 !.
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:
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=
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
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
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
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)
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 =
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
Example 1
Example 1
What is the graph of this function?
Example 1
What is the graph of this function?
f (x) =x2 − 2
x + 2
Example 1
What is the graph of this function?
f (x) =x2 − 2
x + 2
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!
Example 2
Example 2
limx→1
x3 − 1
x − 1
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1=
Example 2
limx→1
x3 − 1
x − 1
Remember how to factor the numerator?
limx→1
x3 − 1
x − 1= lim
x→1
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
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
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
)
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
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
Example 3
Example 3
limh→0
(a+ h)3 − a3
h
Example 3
limh→0
(a+ h)3 − a3
h
Here a is a constant. Let’s expand (a+ h)3:
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
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=
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
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
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
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
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
)
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
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
