day 2: basic properties of limits

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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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Page 1: Day 2: Basic Properties of Limits
Page 2: Day 2: Basic Properties of Limits

Basic Limits

Let’s consider the function f (x) = x .

Page 3: Day 2: Basic Properties of Limits

Basic Limits

Let’s consider the function f (x) = x .Its graphic is:

Page 4: Day 2: Basic Properties of Limits

Basic Limits

Let’s consider the function f (x) = x .Its graphic is:

Page 5: Day 2: Basic Properties of Limits

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

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

Basic Limits

Another basic limit is the limit of a constant:

Page 8: Day 2: Basic Properties of Limits

Basic Limits

Another basic limit is the limit of a constant:

limx→a

b = b

Page 9: Day 2: Basic Properties of Limits

Limits of a Sum

Page 10: Day 2: Basic Properties of Limits

Limits of a Sum

Let’s consider two functions:

Page 11: Day 2: Basic Properties of Limits

Limits of a Sum

Let’s consider two functions:

f (x) = x , g(x) = 5

Page 12: Day 2: Basic Properties of Limits

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

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

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+

Page 15: Day 2: Basic Properties of Limits

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

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

Limits of a Product

Page 18: Day 2: Basic Properties of Limits

Limits of a Product

Let’s consider the function:

Page 19: Day 2: Basic Properties of Limits

Limits of a Product

Let’s consider the function:

f (x) = 3x2

Page 20: Day 2: Basic Properties of Limits

Limits of a Product

Let’s consider the function:

f (x) = 3x2

limx→a

3x2 =

Page 21: Day 2: Basic Properties of Limits

Limits of a Product

Let’s consider the function:

f (x) = 3x2

limx→a

3x2 = limx→a

3.

Page 22: Day 2: Basic Properties of Limits

Limits of a Product

Let’s consider the function:

f (x) = 3x2

limx→a

3x2 = limx→a

3. limx→a

x2

Page 23: Day 2: Basic Properties of Limits

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

Limits of a Quotient

Page 25: Day 2: Basic Properties of Limits

Limits of a Quotient

limx→1

6x + 4

3x − 1=

Page 26: Day 2: Basic Properties of Limits

Limits of a Quotient

limx→1

6x + 4

3x − 1=

limx→1(6x + 4)

limx→1(3x − 1)=

Page 27: Day 2: Basic Properties of Limits

Limits of a Quotient

limx→1

6x + 4

3x − 1=

limx→1(6x + 4)

limx→1(3x − 1)=

=6.1 + 4

3.1 − 1=

Page 28: Day 2: Basic Properties of Limits

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

Example 1

limx→0

(x − 2)(x + 3)

Page 30: Day 2: Basic Properties of Limits

Example 1

limx→0

(x − 2)(x + 3)

limx→0

(x − 2)(x + 3) = limx→0

(x − 2).

Page 31: Day 2: Basic Properties of Limits

Example 1

limx→0

(x − 2)(x + 3)

limx→0

(x − 2)(x + 3) = limx→0

(x − 2). limx→0

(x + 3)

Page 32: Day 2: Basic Properties of Limits

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)

Page 33: Day 2: Basic Properties of Limits

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

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 35: Day 2: Basic Properties of Limits

Example 2

limx→−2

2 − x

x + 1

Page 36: Day 2: Basic Properties of Limits

Example 2

limx→−2

2 − x

x + 1

limx→−2

2 − x

x + 1

Page 37: Day 2: Basic Properties of Limits

Example 2

limx→−2

2 − x

x + 1

limx→−2

2 − x

x + 1=

limx→−2 2 − x

limx→−2 x + 1

Page 38: Day 2: Basic Properties of Limits

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

Page 39: Day 2: Basic Properties of Limits

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

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