4.3 The Limit of Indeterminate Forms and L’Hospital’s Rule

1-11. Evaluate the limits of given indeterminate forms.

1. .

  by L'Hospital Rule, applied twice.

2. .

3. .

 By L'Hospital Rule, .

4.

5. .

      By L'Hospital Rule

6. .

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Ans: 2

7. .

8. .

 By L'Hospital Rule

      .

9. .

10. .

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 .

11. .

                  .

12. Let . Use L'Hospital's Theorem to show that .

               , since is constant and the denominator grows without bound.

13. For what values of and is the following equation correct?

      By L'Hospital Rule for a form of type

      and we may take a to be any value.

14. .

  by L'Hospital Rule for a form of type

15. .

16-17. Find the following limit.

16. .

  (by L'Hospita Rule)  or  .

17. .

                        (by L'Hospital Theorem)

      or 

18-27. Find the limit by using L'Hospital's rule. If you cannot apply L'Hospital's rule, explain why and then find the limits by another method.

18.

 , limit does not exists.

19. .

20. .

 Apply L'Hospital's rule,

 21. .

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22. .

23. .

24. .

  does not exist.

25. .

26. .

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0

27. .

28. The function is defined by

 and .

    Answer the following questions.

    (a) Find .

    (b) Does exist?

    (a)

    (b)

       does not exist.

29. Let be a continuous function with and .

    Find .

 .

30. Let be a angle of a sector of a circle. Find where and are the area of the segment between the chord and the area of the triangle respectively.

 ,