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.

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

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

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

5. Basic Limits
Let’s consider the function f (x) = x.
Its graphic is:
If we fix any value a, we have that:

6. Basic Limits
Let’s consider the function f (x) = x.
Its graphic is:
If we fix any value a, we have that:
lim
x→a
x = a

7. Basic Limits
Another basic limit is the limit of a constant:

8. Basic Limits
Another basic limit is the limit of a constant:
lim
x→a
b = b

9. Limits of a Sum

10. Limits of a Sum
Let’s consider two functions:

11. Limits of a Sum
Let’s consider two functions:
f (x) = x, g(x) = 5

12. Limits of a Sum
Let’s consider two functions:
f (x) = x, g(x) = 5
lim
x→a
[f (x) + g(x)] = lim
x→a
(x + 6)?

13. Limits of a Sum
Let’s consider two functions:
f (x) = x, g(x) = 5
lim
x→a
[f (x) + g(x)] = lim
x→a
(x + 6)?
lim
x→a
(x + 6) =

14. Limits of a Sum
Let’s consider two functions:
f (x) = x, g(x) = 5
lim
x→a
[f (x) + g(x)] = lim
x→a
(x + 6)?
lim
x→a
(x + 6) = lim
x→a
x+

15. Limits of a Sum
Let’s consider two functions:
f (x) = x, g(x) = 5
lim
x→a
[f (x) + g(x)] = lim
x→a
(x + 6)?
lim
x→a
(x + 6) = lim
x→a
x + lim
x→a
6

16. Limits of a Sum
Let’s consider two functions:
f (x) = x, g(x) = 5
lim
x→a
[f (x) + g(x)] = lim
x→a
(x + 6)?
lim
x→a
(x + 6) = lim
x→a
x + lim
x→a
6 = a + 6

17. Limits of a Product

18. Limits of a Product
Let’s consider the function:

19. Limits of a Product
Let’s consider the function:
f (x) = 3x2

20. Limits of a Product
Let’s consider the function:
f (x) = 3x2
lim
x→a
3x2
=

21. Limits of a Product
Let’s consider the function:
f (x) = 3x2
lim
x→a
3x2
= lim
x→a
3.

22. Limits of a Product
Let’s consider the function:
f (x) = 3x2
lim
x→a
3x2
= lim
x→a
3. lim
x→a
x2

23. Limits of a Product
Let’s consider the function:
f (x) = 3x2
lim
x→a
3x2
= lim
x→a
3. lim
x→a
x2
= 3a2

24. Limits of a Quotient

25. Limits of a Quotient
lim
x→1
6x + 4
3x − 1
=

26. Limits of a Quotient
lim
x→1
6x + 4
3x − 1
=
limx→1(6x + 4)
limx→1(3x − 1)
=

27. Limits of a Quotient
lim
x→1
6x + 4
3x − 1
=
limx→1(6x + 4)
limx→1(3x − 1)
=
=
6.1 + 4
3.1 − 1
=

28. Limits of a Quotient
lim
x→1
6x + 4
3x − 1
=
limx→1(6x + 4)
limx→1(3x − 1)
=
=
6.1 + 4
3.1 − 1
=
10
2
= 5.

29. Example 1
lim
x→0
(x − 2)(x + 3)

30. Example 1
lim
x→0
(x − 2)(x + 3)
lim
x→0
(x − 2)(x + 3) = lim
x→0
(x − 2).

31. Example 1
lim
x→0
(x − 2)(x + 3)
lim
x→0
(x − 2)(x + 3) = lim
x→0
(x − 2). lim
x→0
(x + 3)

32. Example 1
lim
x→0
(x − 2)(x + 3)
lim
x→0
(x − 2)(x + 3) = lim
x→0
(x − 2). lim
x→0
(x + 3)
= (0 − 2).(0 + 3)

33. Example 1
lim
x→0
(x − 2)(x + 3)
lim
x→0
(x − 2)(x + 3) = lim
x→0
(x − 2). lim
x→0
(x + 3)
= (0 − 2).(0 + 3) = −6

34. Example 1
lim
x→0
(x − 2)(x + 3)
lim
x→0
(x − 2)(x + 3) = lim
x→0
(x − 2). lim
x→0
(x + 3)
= (0 − 2).(0 + 3) = −6

35. Example 2
lim
x→−2
2 − x
x + 1

36. Example 2
lim
x→−2
2 − x
x + 1
lim
x→−2
2 − x
x + 1

37. Example 2
lim
x→−2
2 − x
x + 1
lim
x→−2
2 − x
x + 1
=
limx→−2 2 − x
limx→−2 x + 1

38. Example 2
lim
x→−2
2 − x
x + 1
lim
x→−2
2 − x
x + 1
=
limx→−2 2 − x
limx→−2 x + 1
=
2 − (−2)
−2 + 1

39. Example 2
lim
x→−2
2 − x
x + 1
lim
x→−2
2 − x
x + 1
=
limx→−2 2 − x
limx→−2 x + 1
=
2 − (−2)
−2 + 1
= −4

40. Example 2
lim
x→−2
2 − x
x + 1
lim
x→−2
2 − x
x + 1
=
limx→−2 2 − x
limx→−2 x + 1
=
2 − (−2)
−2 + 1
= −4