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. .

http://matrix.skku.ac.kr/cal-lab/cal-4-3-6.html

 


Ans: 2


7. .

 


8. .

 By L'Hospital Rule

      .


9. .

 


10. .

http://matrix.skku.ac.kr/cal-lab/cal-4-3-10.html

 .


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. .

http://matrix.skku.ac.kr/cal-lab/cal-4-3-19.html

 


22. .

 


23. .

 


24. .

  does not exist.


25. .

 


26. .

http://matrix.skku.ac.kr/cal-lab/cal-4-3-24.html (changed)

 

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.

 ,