Lect

1. lecture 2
Prepared by:
Assist. Lect. Hiba Abdul –Kareem Assist. Lect. Azam Isam

2. The Limits

3. Properties of Limits
1. 1-Limit of a constant:
Example :
𝐥𝐢𝐦
𝒙→𝒄
(𝟕)= 7
𝐥𝐢𝐦
𝒙→𝟖
(𝟕)= 7

4. Properties of Limits
2- Constant Multiplied by a Function (Constant Multiple
Rule)
Example :
lim
𝑥→2
5 𝑥 2=20

5. Properties of Limits
2- Constant Multiplied by a Function (Constant Multiple
Rule)
Example :
lim
𝑥→2
5 𝑥 2=20

6. Properties of Limits
3. Sum of Functions
The limit of a sum equals the sum of the limits. In other words,
figure out the limit for each piece, then add them together.
Example :

7. Properties of Limits
4. Product of Two or More Functions
Example :

8. Properties of Limits
5. Quotient of Two or More Functions
Example :

9. Properties of Limits
6.Evaluating the Limit of the Power of a Function
Example :

10. How to solve limits
How to solve
limits
Multiplying
By Conjugate
Analysis
trigonometric
identities

11. How to solve limits : Analysis

12. How to solve limits : Analysis
Example

13. How to solve limits : Analysis
Example

14. How to solve limits : Analysis
Example
lim
𝑥→2
𝑥2
− 4
𝑥 − 2
= lim
𝑥→2
𝑥−2 ⋅ 𝑥+2
𝑥−2
=4
Example
lim
𝑥→−2
𝑥3
+ 8
𝑥 + 2
=
lim
𝑥→−2
𝑥+2 𝑥2−2𝑥+4
𝑥+2
=12

15. How to solve limits : Analysis
Example

16. How to solve limits : Analysis
Example

17. How to solve limits : Analysis
Example

18. Homework
1. lim
𝑥→3
𝑥−3
𝑥2+𝑥−12
2. lim
𝑥→4
8−2𝑥
𝑥2−𝑥−12

19. How to solve limits : trigonometric
identities

20. How to solve limits : trigonometric
identities

21. How to solve limits : trigonometric
identities
Example
lim
𝑥→
𝜋
2
sin 2𝑥
cos − 𝑥
= lim
𝑥→
𝜋
2
2 sin 𝑥 ⋅ cos 𝑥
cos 𝑥
= 2
Example
lim
𝑥→0
sin2
𝑥
1 − cos 2𝑥
= lim
𝑥→0
sin2 𝑥
1 − 1 − 2 sin2 𝑥
= lim
𝑥→0
1
2
= 2

22. How to solve limits : trigonometric

23. How to solve limits
• Multiplying By Conjugate
Example

24. How to solve limits
• Multiplying By Conjugate
Example

25. L'Hopital's rule

26. Apply l'Hôpital's rule:
L'Hôpital's rule
Example

27. L'Hôpital's rule
Example

28. L'Hôpital's rule
Example

29. L'Hôpital's rule
Example

30. L'Hôpital's rule
Homework
lim
𝑥→−2
𝒙 + 𝟏𝟎 + 3𝑥1/3
4𝑥2 + 3𝑥 − 1

31. Any
Questions