1. 1 羅必達法則 (L ’ Hospital ’ s Rule) 1. 不定式 (Indeterminate Forms) 2. 羅必達定理 (L’Hopital’s Rule) 3. 例題 page 659-663

2. 2 Indeterminate Forms 1. The Indeterminate Forms of Type 2. The Indeterminate Forms of Type 3. The Indeterminate Forms and 4. The Indeterminate Forms, and EX:

3. 3 The Indeterminate Forms of Type 0/0 Take for example When  &  Divide both numerator and denominator by x-1

4. 4 The Indeterminate Forms of Type 0/0

5. 5 Replace by Replace x −1 by if,, exist and, then the weak form of L’Hopital’s Rule

6. 6 L ’ Hospital ’ s Rule Let f and g be functions and let a be a real number such that Let f and g have derivative that exist at each point in some open interval containing a If, then If does exist because becomes large without bound for values of x near a, then also does not exist

7. 7 EX1 L ’ Hospital ’ s Rule Find Check the conditions of L’Hospital’s Rule If then f’ ( x )=2 x If f ( x )= x -1 then f’ ( x )=1 By L’Hospital’s Rule, this result is the desired limit:

8. 8 EX2 L ’ Hospital ’ s Rule Find Check the conditions of L’Hospital’s Rule If then f’ ( x )= If f ( x )= then f’ ( x )=2( x -1) Because does not exist Then does not exist

9. 9 Using L ’ Hospital ’ s Rule 1. Be sure that leads to the indeterminate form 0/0. 2. Take the derivates of f and g seperately. 3. Find the limit of ; this limit, if it exists, equals the limit of f(x)/g(x). 4. If necessary, apply L’Hospital’s rule more than once.

10. 10 EX3 L ’ Hospital ’ s Rule Find Check the conditions of L’Hospital’s Rule If then f’ ( x )= If f ( x )= then f’ ( x )=

11. 11 EX4-1 L ’ Hospital ’ s Rule Find If then f’ ( x )= If f ( x )= then f’ ( x )=2 x 

12. 12 EX4-2 L ’ Hospital ’ s Rule If then f’ ( x )= If f ( x )= then f’ ( x )=2

13. 13 EX5 L ’ Hospital ’ s Rule Find  (by substitution)

14. 14 Proof of L ’ Hospital ’ s Rule-1 We can prove the theorem for special case f, g, f’, g’ are continuous on some open interval containing a, and g’(a)= 0. With these assumptions the fact that and means that both f(a)=0 and g(a)= 0

15. 15 Proof of L ’ Hospital ’ s Rule-2 Thus, Multiplying the numerator and denominator by 1/( x-a ) gives

16. 16 Proof of L ’ Hospital ’ s Rule-3 By the property of limits, this becomes, the limit of numerator is f’(a) the limit of denominator is g’(a) and

17. 17 Proof of L ’ Hospital ’ s Rule-4 Thus,

18. 18 Example: Find (0/0)

19. 19 Example: Find (0/0)

20. 20 Example: Find (0/0)

21. 21 The Indeterminate Forms of Type If and Then

22. 22 Example (∞/∞) Find

23. 23 Example: Find, where p>0 。

24. 24 Example: Find ( ∞/∞ )

25. 25 Example: Find (∞/∞)

26. 26 Example: Find (a>0) (∞/∞)

27. 27 Example: Find (∞/∞)

28. 28 Example: Find (∞/∞)

29. 29 The Indeterminate Forms and To evaluate Rewrite Or Then apply L’Hospital’s Rule

30. 30 The Indeterminate Forms and To evaluate F(x)-g(x) must rewrite as a single term. When the trigonometric functions are involved, switching to all sines and cosins may help.

31. 31 Example: Find

32. 32 Example: Find

33. 33 Example: Find

34. 34 Example: Find (∞−∞)

35. 35 Example: Find (∞−∞)

36. 36 Example: Find (∞−∞)

37. 37 Example: Find

38. 38 Example: Find

39. 39 Example: Find

40. 40 The Indeterminate Forms, and In these cases 1. Let 2. 3. If exists and equal L, then

41. 41 Example: Find

42. 42 Example: Find and Then

43. 43 Example: Find

44. 44 Then

45. 45 Example: 。

46. 46 Example: Find

47. 47 Example: Find Replace the result of

48. 48 Then

49. 49 Example:

50. 50 and Then Example: