Chapter 12. LIMITS & DERIVATIVES

Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1 , Exercise 12.2 and Miscellaneous Exercise Questions and Solutions :

Class 11 Maths Chapter 12. Limits and Derivatives Exercise 12.1 Questions and Solutions :

Evaluate the following limits in Exercises 1 to 22 .

1.      

Solution :  We have,

   

2. 

Solution : We have,

 

3.

Solution : We have,

 

4. 

Solution : We have,

 

5.

Solution : We have,

6. 

Solution : We have, 

Let when ,then

 

7.

Solution : We have,

8.

Solution : We have, 

9. 

Solution : We have,

10. 

Solution: We have,

 

11. 

Solution: We have,

12. 

Solution: We have, 

 

13. 

Solution : We have, 

14.

Solution: We have,

15. 

Solution : We have,

Let  when , then

16.

Solution : We have ,

17.

Solution:  We have, 

18. 

Solution : We have, 

19.

Solution : We have,

20.

Solution : We have , 

21.

Solution : We have,  

22. 

Solution : We have,  

Let when , then

23. Find   and  , where     .

Solution : Given, the function     

At

 

Let  when  then  

    

Let  when  then  

 

Therefore, the function is exist at  .

At     

Let  when  then  

    

Let  when  then  

Therefore, the function is exist at  .

24. Find  , where          

Solution :   Given , the function          

At      

Let  when  then  

     

Let  when  then  

 

Therefore, the function does not exist at  .

25. Evaluate   , where   .

Solution : Given, the function 

At    

Let  when  then  

Let  when  then  

Therefore, the function does not exist at  .

26. Find , where .

Solution :  Given, the function 

At   

Let  when  then  

Let  when  then  

Therefore, the function does not exist at  .

27. Find   , where   .

Solution : Given, the function     

   

 

28. Suppose    and if   what are possible values of  and ?

Solution : Given, the function  

                            

At      

Let  when  then  

    

Let  when  then  

So,  

 and  

 

 

29. Let  be fixed real numbers and define a function   . What is  ? For some   . compute  .        

Solution :  Given, the function

Again ,  

30. If   . For What values of  does   exists ?

Solution :  Given , the function  

At   

Let  when  then  

Let  when  then  

So , the function does not exist at  .

Therefore,  is exist for all the value of  , except  .

31. If the function  satisfies  , evaluate   .

Solution :   We have,

      

   

 

 

32. If     . For what integers  and  does both  and  exist ?

Solution :  Given , the function     

Let  when  then  

     

Let  when  then  

Therefore, the function is exist at  .

 

      

At       

Let  when  then  

     

Let  when  then  

  

  

Therefore, the function is exist at  .

  

       

Class 11 Maths Chapter 13. Limits and Derivatives Exercise 13.2 Questions and Solutions :

1. Find the derivative of  at   .   

Solution : let   

By definition of the derivative function, we have

At ,                  

2. Find the derivative of  at   .  

Solution : let  

By definition of the derivative function, we have

At ,                                       

3. Find the derivative of  at  .     

Solution : let   

By definition of the derivative function, we have

At ,                                      

4. Find the derivative of the following functions from first principle .

    (i)      (ii)      (iii)       (iv)       

Solution : (i)  let  

By definition of the derivative function, we have

  

Solution :  (ii) let

By definition of the derivative function, we have

Solution :  (ii) let 

By definition of the derivative function, we have

Solution :  (iii)  Let 

By definition of the derivative function, we have

 

5. For the function :  . Prove that  .

Solution :  [Note : If  , then    ]

We have ,

If  ,  then

 

If  , then

 (100 times)

 Proved .

6. Find the derivative of  for some fixed real number  .

Solution : let 

Differentiating with respect to x , we have

7. For some constants  and , find the derivative of

(i)     (ii)     (iii) 

Solution : (i) let 

   

Differentiating with respect to x , we have

   

(ii)  Let

    

Differentiating with respect to x , we have

Solution : (iii)  [Note :    ]

 Let 

Differentiating with respect to x , we have

8. Find the derivative of for some constant  .

Solution : let 

Differentiating with respect to x , we have

9. Find the derivative of  (i)      (ii)   (iii)   (iv)    (v)     (vi) 

Solution : (i)

Let 

Differentiating with respect to x , we have

    [Since,   ]

Solution : (ii)  

Let  

Differentiating with respect to x , we have

 

Solution : (iii) let  

Differentiating with respect to x , we have

  

Solution : (iv)  let  

Differentiating with respect to x , we have

Solution : (v) Let    

    

Differentiating with respect to x , we have

    

    

Solution : (vi) let 

Differentiating with respect to x , we have

10. Find the derivative of  from first principle .

Solution : let

By definition of the derivative function, we have

[Note : ]

11.  Find the derivative of the following functions :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)           

[ Formula :       ;      ;     ;    ;     ;   ]

 Solution :  (i) let

Differentiating with respect to x , we have

Solution : (ii) let 

Differentiating with respect to x , we have

 

Solution : (iii) let

Differentiating with respect to x , we have

Solution : (iv) Let

Differentiating with respect to x , we have

 

Solution :  (v) let

Differentiating with respect to x , we have

Solution :  (vi) let

Differentiating with respect to x , we have

 

 

(vii) let     

Differentiating with respect to x , we have

Class 11 Maths Chapter 13 Limits and Derivatives Miscellaneous Exercise Questions and Solutions :

1. Find the derivative of the following functions from first principle :

(i)     (ii)      (iii)      (iv)

Solution : (i) Let

    

By definition of the derivative function, we have

Solution : (ii) let 

By definition of the derivative function, we have

 Solution : (iii) Let      

By definition of the derivative function, we have

[ Note : ]

Solution : (iv) Let

By definition of the derivative function, we have

[Note : ]

 Find the derivative of the following functions (it is to be understood that  and  are fixed non-zero constants and  and  are integers) :

2.     

3.

4.

5.

6. 

7.

8.

9.

10.

11.     

12.   

13.  

14.    

15.

16.

17. 

18.

19.

20.

21.

22.  

23.   

24.  

25.

26.

27.

28.

29.

30.  

Solution :   2. Let    

Differentiating with respect to x , we have

 Solution :  3.   Let

Differentiating with respect to x , we have

 

Solution : 4. 

Differentiating with respect to x , we have

Solution : 5.  Let 

Differentiating with respect to x , we have

Solution : 6. Let  

Differentiating with respect to x , we have

Solution : 7. Let 

Differentiating with respect to x , we have

Solution :  8. Let 

Differentiating with respect to x , we have

 

 

 

 

Solution : 9.  Let

Differentiating with respect to x , we have

 

Solutioon : 10. Let 

Differentiating with respect to x , we have

 

Solution : 11. Let     

     

Differentiating with respect to x , we have

 

Solution : 12. Let   

Differentiating with respect to x , we have

Solution : 13. Let   

Differentiating with respect to x , we have

Solution :  14. Let

Differentiating with respect to x , we have

 

 Solution :  15. Let

Differentiating with respect to x , we have

 

 

Solution : 16.  Let 

Differentiating with respect to x , we have

 

Solution : 17. Let 

Differentiating with respect to x , we have

 

 

 

 

 

 

 

 

 

 

Solution : 18. Let

Differentiating with respect to x , we have

Solution : 19. Let 

Differentiating with respect to x , we have

Solution : 20. Let

Differentiating with respect to x , we have

Solution : 21. Let

Differentiating with respect to x , we have

[Note :  ]

Solution : 22.  

Let  

Differentiating with respect to x , we have

Solution : 23. Let     

Differentiating with respect to x , we have

Solution : 24.  

Let

Differentiating with respect to x , we have

Solution : 25.  Let

Differentiating with respect to x , we have

Solution : 26.  Let

Differentiating with respect to x , we have

Solution : 27. Let 

Differentiating with respect to x , we have

Solutiopn : 28. Let 

Differentiating with respect to x , we have

Solution : 29.

Let 

Differentiating with respect to x , we have

Solution : 30.   Let  

Differentiating with respect to x , we have