Calculus-Sec-4-3-Solution

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

by SGLee - HSKim -JHLee

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

1.

Apply L'Hospital Rule twice.

2.

3.

By L'Hospital Rule, .

4.

5.

Let .  Note  and

. By L`Hospital’s Rule,

()

.

7.

1

8.

By L'Hospital’s Rule

.

9.

11.

.

12. Let . Use L'Hospital's Theorem to show that .     <-- x가 +infinity로 가야 함 (수정 필요)

, since  is constant and the denominator grows without bound.

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

By L'Hospital’s Rule for a form of type

and we may take  to be any value.

14.

by L'Hospital Rule for a form of type

15.

. Note
Therefore, by L’Hospital’s Rule,

.

16-17. Find the following limit.

16.

(by L'Hospital’s Rule)  or  .

17.

(by L'Hospital’s 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 limit by another method.

18.

, limit does not exists.

19.

(by L`Hospital`s Rule)  .

20.

Apply L'Hospital's rule,

22.

23.

24.

does not exist.

25.

27.

28. The function  is defined by

and  .

(a) Find .

(b) Does  exist?

(a)

(b)

does not exist.

29. Let  be a continuous function with  and . Find .

and . Hence, by L'Hospital’s Theorem

.

30. Let  be the angle of a sector of a circle. Find  where  and  are

the area of the segment shaded region between the chord  and arc, the area of the triangle  respectively.

,

.