羅必達法則 (L ’ Hospital ’ s Rule)

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. 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. The Indeterminate Forms of Type0/0 Take for example When  &  Divide both numerator and denominator by x-1

4. The Indeterminate Forms of Type0/0

5. The Indeterminate Forms of Type0/0 Replace by Replace by Replace x−1 by if , , exist and , then the weak form of L’Hopital’s Rule

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

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

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

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

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

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. Proof of L’Hospital’s Rule-4 Thus,

18. Example: Find (0/0)

19. Example: Find (0/0)

20. Example: Find (0/0)

21. The Indeterminate Forms of Type If and Then

22. Example(∞/∞) • Find

23. Example: Find , where p>0。

24. Example: Find (∞/∞)

25. Example: Find (∞/∞)

26. Example: Find (a>0) (∞/∞)

27. Example: Find (∞/∞)

28. Example: Find (∞/∞)

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

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. Example: Find

32. Example: Find

33. Example: Find

34. Example: Find (∞−∞)

35. Example: Find (∞−∞)

36. Example: Find (∞−∞)

37. Example: Find

38. Example: Find

39. Example: Find

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

41. Example: Find

42. Example: Find and Then

43. Example: Find

44. Then

45. Example: 。

46. Example: Find

47. Example: Find Replace the result of

48. Then

49. Example:

50. Example: and Then