day 2: basic properties of limits
DESCRIPTION
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.TRANSCRIPT
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Basic Limits
Let’s consider the function f (x) = x .
![Page 3: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/3.jpg)
Basic Limits
Let’s consider the function f (x) = x .Its graphic is:
![Page 4: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/4.jpg)
Basic Limits
Let’s consider the function f (x) = x .Its graphic is:
![Page 5: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/5.jpg)
Basic Limits
Let’s consider the function f (x) = x .Its graphic is:
If we fix any value a, we have that:
![Page 6: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/6.jpg)
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
![Page 7: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/7.jpg)
Basic Limits
Another basic limit is the limit of a constant:
![Page 8: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/8.jpg)
Basic Limits
Another basic limit is the limit of a constant:
limx→a
b = b
![Page 9: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/9.jpg)
Limits of a Sum
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Limits of a Sum
Let’s consider two functions:
![Page 11: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/11.jpg)
Limits of a Sum
Let’s consider two functions:
f (x) = x , g(x) = 5
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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)?
![Page 13: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/13.jpg)
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) =
![Page 14: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/14.jpg)
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+
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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
![Page 16: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/16.jpg)
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
![Page 17: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/17.jpg)
Limits of a Product
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Limits of a Product
Let’s consider the function:
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Limits of a Product
Let’s consider the function:
f (x) = 3x2
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Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 =
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Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 = limx→a
3.
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Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 = limx→a
3. limx→a
x2
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Limits of a Product
Let’s consider the function:
f (x) = 3x2
limx→a
3x2 = limx→a
3. limx→a
x2 = 3a2
![Page 24: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/24.jpg)
Limits of a Quotient
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Limits of a Quotient
limx→1
6x + 4
3x − 1=
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Limits of a Quotient
limx→1
6x + 4
3x − 1=
limx→1(6x + 4)
limx→1(3x − 1)=
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Limits of a Quotient
limx→1
6x + 4
3x − 1=
limx→1(6x + 4)
limx→1(3x − 1)=
=6.1 + 4
3.1 − 1=
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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.
![Page 29: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/29.jpg)
Example 1
limx→0
(x − 2)(x + 3)
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Example 1
limx→0
(x − 2)(x + 3)
limx→0
(x − 2)(x + 3) = limx→0
(x − 2).
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Example 1
limx→0
(x − 2)(x + 3)
limx→0
(x − 2)(x + 3) = limx→0
(x − 2). limx→0
(x + 3)
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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)
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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
![Page 34: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/34.jpg)
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
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Example 2
limx→−2
2 − x
x + 1
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Example 2
limx→−2
2 − x
x + 1
limx→−2
2 − x
x + 1
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Example 2
limx→−2
2 − x
x + 1
limx→−2
2 − x
x + 1=
limx→−2 2 − x
limx→−2 x + 1
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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
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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
![Page 40: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/40.jpg)
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
![Page 41: Day 2: Basic Properties of Limits](https://reader036.vdocument.in/reader036/viewer/2022081401/55857351d8b42a3d2c8b4c9f/html5/thumbnails/41.jpg)