Day 2: Basic Properties of Limits
Jun 20, 2015
In this second day we solve the most basic limits we could find, like the limit of a constant. Then we find the limit of the sum, the product and the quotient of two functions. We solve two simple examples.
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Basic Limits
Let’s consider the function f (x) = x .
Basic Limits
Let’s consider the function f (x) = x .Its graphic is:
Basic Limits
Let’s consider the function f (x) = x .Its graphic is:
Basic Limits
Let’s consider the function f (x) = x .Its graphic is:
If we fix any value a, we have that:
Basic Limits
Let’s consider the function f (x) = x .Its graphic is:
If we fix any value a, we have that:
limx→a
x = a
Basic Limits
Another basic limit is the limit of a constant:
Basic Limits
Another basic limit is the limit of a constant:
limx→a
b = b
Limits of a Sum
Limits of a Sum
Let’s consider two functions:
Limits of a Sum
Let’s consider two functions:
f (x) = x , g(x) = 5
Limits of a Sum
Let’s consider two functions:
f (x) = x , g(x) = 5
limx→a
[f (x) + g(x)] = limx→a
(x + 6)?
Limits of a Sum
Let’s consider two functions:
f (x) = x , g(x) = 5
limx→a
[f (x) + g(x)] = limx→a
(x + 6)?
limx→a
(x + 6) =
Limits of a Sum
Let’s consider two functions:
f (x) = x , g(x) = 5
limx→a
[f (x) + g(x)] = limx→a
(x + 6)?
limx→a
(x + 6) = limx→a
x+
Limits of a Sum
Let’s consider two functions:
f (x) = x , g(x) = 5
limx→a
[f (x) + g(x)] = limx→a
(x + 6)?
limx→a
(x + 6) = limx→a
x + limx→a
6
Limits of a Sum
Let’s consider two functions:
f (x) = x , g(x) = 5
limx→a
[f (x) + g(x)] = limx→a
(x + 6)?
limx→a
(x + 6) = limx→a
x + limx→a
6 = a + 6
Limits of a Product
Limits of a Product
Let’s consider the function:
Limits of a Product
Let’s consider the function:
f (x) = 3x2
Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 =
Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 = limx→a
3.
Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 = limx→a
3. limx→a
x2
Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 = limx→a
3. limx→a
x2 = 3a2
Limits of a Quotient
Limits of a Quotient
limx→1
6x + 4
3x − 1=
Limits of a Quotient
limx→1
6x + 4
3x − 1=
limx→1(6x + 4)
limx→1(3x − 1)=
Limits of a Quotient
limx→1
6x + 4
3x − 1=
limx→1(6x + 4)
limx→1(3x − 1)=
=6.1 + 4
3.1 − 1=
Limits of a Quotient
limx→1
6x + 4
3x − 1=
limx→1(6x + 4)
limx→1(3x − 1)=
=6.1 + 4
3.1 − 1=
10
2= 5.
Example 1
limx→0
(x − 2)(x + 3)
Example 1
limx→0
(x − 2)(x + 3)
limx→0
(x − 2)(x + 3) = limx→0
(x − 2).
Example 1
limx→0
(x − 2)(x + 3)
limx→0
(x − 2)(x + 3) = limx→0
(x − 2). limx→0
(x + 3)
Example 1
limx→0
(x − 2)(x + 3)
limx→0
(x − 2)(x + 3) = limx→0
(x − 2). limx→0
(x + 3)
= (0 − 2).(0 + 3)
Example 1
limx→0
(x − 2)(x + 3)
limx→0
(x − 2)(x + 3) = limx→0
(x − 2). limx→0
(x + 3)
= (0 − 2).(0 + 3) = −6
Example 1
limx→0
(x − 2)(x + 3)
limx→0
(x − 2)(x + 3) = limx→0
(x − 2). limx→0
(x + 3)
= (0 − 2).(0 + 3) = −6
Example 2
limx→−2
2 − x
x + 1
Example 2
limx→−2
2 − x
x + 1
limx→−2
2 − x
x + 1
Example 2
limx→−2
2 − x
x + 1
limx→−2
2 − x
x + 1=
limx→−2 2 − x
limx→−2 x + 1
Example 2
limx→−2
2 − x
x + 1
limx→−2
2 − x
x + 1=
limx→−2 2 − x
limx→−2 x + 1
=2 − (−2)
−2 + 1
Example 2
limx→−2
2 − x
x + 1
limx→−2
2 − x
x + 1=
limx→−2 2 − x
limx→−2 x + 1
=2 − (−2)
−2 + 1= −4
Example 2
limx→−2
2 − x
x + 1
limx→−2
2 − x
x + 1=
limx→−2 2 − x
limx→−2 x + 1
=2 − (−2)
−2 + 1= −4
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