Limit Continuity and Derivative Solved Examples | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Solved Example on Limit Continuity and 
Derivative 
JEE Mains 
Q1:  If ?? (?? +?? )=?? (?? )·?? (?? ) for all ?? and ?? and ?? (?? )=?? ,?? '
(?? )=?? , then ?? '
(?? ) will be 
(a) 2 
(b) 4 
(c) 6 
(d) 8 
Ans: (c) Let ?? =5,?? =0??? (5+0)=?? (5)·?? (0) 
??? (5)=?? (5)?? (0)??? (0)=1 
Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] 
 Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] {??? (5)=2}
 =2lim
h?0
?[
?? (h)-?? (0)
h
]=2×?? '
(0)=2×3=6.
 
Q2:  If ?? (?? )=?? ,?? '
(?? )=-?? ,?? (?? )=-?? ,?? '
(?? )=?? , then ?????? ?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = 
(a) -5 
(b) 10 
(c) -10 
(d) 5 
Ans: (b) lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? . We add and subtract ?? (?? )?? (?? ) in numerator 
 = lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )+?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]- lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]
 = ?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ]-?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ] = ?? (?? )?? '
(?? )-?? (?? )?? '
(?? )  [by using first principle formula] 
 = 3.4-(-1)(-2)= 12-2= 10
 
[by using first principle formula] 
Trick : lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? 
Using L-Hospital's rule, Limit =lim
?? ??? ?
?? '
(?? )?? (?? )-?? (?? )?? '
(?? )
1
; 
Limit =?? '
(?? )?? (?? )-?? (?? )?? '
(?? )=(4)(3)-(-1)(-2)=12-2=10. 
 
Page 2


Solved Example on Limit Continuity and 
Derivative 
JEE Mains 
Q1:  If ?? (?? +?? )=?? (?? )·?? (?? ) for all ?? and ?? and ?? (?? )=?? ,?? '
(?? )=?? , then ?? '
(?? ) will be 
(a) 2 
(b) 4 
(c) 6 
(d) 8 
Ans: (c) Let ?? =5,?? =0??? (5+0)=?? (5)·?? (0) 
??? (5)=?? (5)?? (0)??? (0)=1 
Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] 
 Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] {??? (5)=2}
 =2lim
h?0
?[
?? (h)-?? (0)
h
]=2×?? '
(0)=2×3=6.
 
Q2:  If ?? (?? )=?? ,?? '
(?? )=-?? ,?? (?? )=-?? ,?? '
(?? )=?? , then ?????? ?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = 
(a) -5 
(b) 10 
(c) -10 
(d) 5 
Ans: (b) lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? . We add and subtract ?? (?? )?? (?? ) in numerator 
 = lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )+?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]- lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]
 = ?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ]-?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ] = ?? (?? )?? '
(?? )-?? (?? )?? '
(?? )  [by using first principle formula] 
 = 3.4-(-1)(-2)= 12-2= 10
 
[by using first principle formula] 
Trick : lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? 
Using L-Hospital's rule, Limit =lim
?? ??? ?
?? '
(?? )?? (?? )-?? (?? )?? '
(?? )
1
; 
Limit =?? '
(?? )?? (?? )-?? (?? )?? '
(?? )=(4)(3)-(-1)(-2)=12-2=10. 
 
Q3:  If ?? ?? (?? )+?? ?? (
?? ?? )=?? +?? and ?? =???? (?? ) then (
????
????
)
?? =?? is equal to 
(a) 14 
(b) 
?? ?? 
(c) 1 
(d) None of these 
Ans: (b)  ?5?? (?? )+3?? (
1
?? )=?? +2   
Replacing ?? by 
1
?? in (i), 5?? (
1
?? )+3?? (?? )=
1
?? +2 
On solving equation (i) and (ii), we get, 16?? (?? )=5?? -
3
?? +4,?16?? '
(?? )=5+
3
?? 2
 
 ??? =???? (?? )?
????
????
=?? (?? )+?? ?? '
(?? )=
1
16
(5?? -
3
?? +4)+?? ·
1
16
(5+
3
?? 2
)
 at ?? =1,
????
????
=
1
16
(5-3+4)+
1
16
(5+3)=
7
8
.
 
Q4: The derivative of ?? (?? )=?? |
?? at ?? =?? is 
(a) ?? 
(b) 1 
(c) -1 
(d) Not defined 
Ans:   (a)  ?? (?? )={
?? 3
,?? =0
-?? 3
, ?? <0
  and  ?? '
(?? )={
3?? 2
,?? =0
-3?? 2
,?? <0
 
 
?? '
(0
+
)=?? '
(0
-
)=0 
 
Q5: Example: 6 The first derivative of the function (?????? ?? ?? ?????? ?? ?? ?????? ?? ?? +??????
?? ?? ?? +?? ) with 
respect to ?? at ?? =?? is 
(a) 2 
(b) -1 
(c) -?? +?? ?? ?????? ?? ?? 
(d) -?? +??????
?? ?? 
Ans: (b)  ?? (?? )=sin 2?? ·cos 2?? ·cos 3?? +log
2
 2
?? +3
, ?? (?? )=
1
2
sin 4?? cos 3?? +(?? +
3)log
2
 2, ?? (?? )=
1
4
[sin 7?? +sin ?? ]+?? +3 
Differentiate w.r.t. ?? , 
?? '
(?? )=
1
4
[7cos 7?? +cos ?? ]+1,?? '
(?? )=
1
4
7cos 7?? +
1
4
cos ?? +1, ?? '
(?? )=-2+1=-1. 
 
 
 
Page 3


Solved Example on Limit Continuity and 
Derivative 
JEE Mains 
Q1:  If ?? (?? +?? )=?? (?? )·?? (?? ) for all ?? and ?? and ?? (?? )=?? ,?? '
(?? )=?? , then ?? '
(?? ) will be 
(a) 2 
(b) 4 
(c) 6 
(d) 8 
Ans: (c) Let ?? =5,?? =0??? (5+0)=?? (5)·?? (0) 
??? (5)=?? (5)?? (0)??? (0)=1 
Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] 
 Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] {??? (5)=2}
 =2lim
h?0
?[
?? (h)-?? (0)
h
]=2×?? '
(0)=2×3=6.
 
Q2:  If ?? (?? )=?? ,?? '
(?? )=-?? ,?? (?? )=-?? ,?? '
(?? )=?? , then ?????? ?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = 
(a) -5 
(b) 10 
(c) -10 
(d) 5 
Ans: (b) lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? . We add and subtract ?? (?? )?? (?? ) in numerator 
 = lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )+?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]- lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]
 = ?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ]-?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ] = ?? (?? )?? '
(?? )-?? (?? )?? '
(?? )  [by using first principle formula] 
 = 3.4-(-1)(-2)= 12-2= 10
 
[by using first principle formula] 
Trick : lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? 
Using L-Hospital's rule, Limit =lim
?? ??? ?
?? '
(?? )?? (?? )-?? (?? )?? '
(?? )
1
; 
Limit =?? '
(?? )?? (?? )-?? (?? )?? '
(?? )=(4)(3)-(-1)(-2)=12-2=10. 
 
Q3:  If ?? ?? (?? )+?? ?? (
?? ?? )=?? +?? and ?? =???? (?? ) then (
????
????
)
?? =?? is equal to 
(a) 14 
(b) 
?? ?? 
(c) 1 
(d) None of these 
Ans: (b)  ?5?? (?? )+3?? (
1
?? )=?? +2   
Replacing ?? by 
1
?? in (i), 5?? (
1
?? )+3?? (?? )=
1
?? +2 
On solving equation (i) and (ii), we get, 16?? (?? )=5?? -
3
?? +4,?16?? '
(?? )=5+
3
?? 2
 
 ??? =???? (?? )?
????
????
=?? (?? )+?? ?? '
(?? )=
1
16
(5?? -
3
?? +4)+?? ·
1
16
(5+
3
?? 2
)
 at ?? =1,
????
????
=
1
16
(5-3+4)+
1
16
(5+3)=
7
8
.
 
Q4: The derivative of ?? (?? )=?? |
?? at ?? =?? is 
(a) ?? 
(b) 1 
(c) -1 
(d) Not defined 
Ans:   (a)  ?? (?? )={
?? 3
,?? =0
-?? 3
, ?? <0
  and  ?? '
(?? )={
3?? 2
,?? =0
-3?? 2
,?? <0
 
 
?? '
(0
+
)=?? '
(0
-
)=0 
 
Q5: Example: 6 The first derivative of the function (?????? ?? ?? ?????? ?? ?? ?????? ?? ?? +??????
?? ?? ?? +?? ) with 
respect to ?? at ?? =?? is 
(a) 2 
(b) -1 
(c) -?? +?? ?? ?????? ?? ?? 
(d) -?? +??????
?? ?? 
Ans: (b)  ?? (?? )=sin 2?? ·cos 2?? ·cos 3?? +log
2
 2
?? +3
, ?? (?? )=
1
2
sin 4?? cos 3?? +(?? +
3)log
2
 2, ?? (?? )=
1
4
[sin 7?? +sin ?? ]+?? +3 
Differentiate w.r.t. ?? , 
?? '
(?? )=
1
4
[7cos 7?? +cos ?? ]+1,?? '
(?? )=
1
4
7cos 7?? +
1
4
cos ?? +1, ?? '
(?? )=-2+1=-1. 
 
 
 
Q6: ?? is a point on the circumference of a circle & ?? is the foot of the perpendicular from ?? on a 
fixed diameter ???? . Then the limit of 
?? ?? ?? ????
 as ?? tends to ?? along the circumference 
(A) Does not exist 
(B) Equal to one 
(C) Is equal to the length ???? 
(D) None 
Ans: (C) 
Hint: ?? ?? 2
=???? ×????  
 
 Q8: ?????? ?? ??? ?(?? +??????
?????? 
?? ?? ?? ?????? ?? ) 
 
(A) Is equal to 4 (B) Is equal to 25 
(C) Is equal to 289 (D) Is non existent 
 
 
Ans: (C) 
 lim
?? ?0
?(1+log
cos 
?? 2
2
 cos ?? )
2
 lim
?? ?0
?log
cos 
?? 2
 cos ?? =lim
?? ?0
?
log (cos ?? )
log (cos 
?? 2
)
 =lim
?? ?0
?
-sin ?? cos ?? 1
2
sin ?? /2
cos ?? /2
=lim
?? ?0
?
4cos
2
 
?? 2
cos ?? =4
 
 
Q9: ?????? ?? ??? ?
(?? +?? )
?? ?? -?? (?? +?? )
?? ?? -?? is 
(A) 1 
(B) 0 
(C) ?? /?? 
(D) 8 
Ans: (C) Use lim
?? ??? ?
?? ?? -?? ?? ?? -?? =?? ?? ?? -1
 
Q10: Centre of circle is the limit of point of intersection of tines ?? ?? +?? ?? =?? and (?? +?? )?? +
?? ?? ?? ?? =?? as ?? tends to 1 . If it passes through (?? ,?? ) its radius is - 
(A) 
v????????
????
 
(B) 
????
????
 
Page 4


Solved Example on Limit Continuity and 
Derivative 
JEE Mains 
Q1:  If ?? (?? +?? )=?? (?? )·?? (?? ) for all ?? and ?? and ?? (?? )=?? ,?? '
(?? )=?? , then ?? '
(?? ) will be 
(a) 2 
(b) 4 
(c) 6 
(d) 8 
Ans: (c) Let ?? =5,?? =0??? (5+0)=?? (5)·?? (0) 
??? (5)=?? (5)?? (0)??? (0)=1 
Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] 
 Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] {??? (5)=2}
 =2lim
h?0
?[
?? (h)-?? (0)
h
]=2×?? '
(0)=2×3=6.
 
Q2:  If ?? (?? )=?? ,?? '
(?? )=-?? ,?? (?? )=-?? ,?? '
(?? )=?? , then ?????? ?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = 
(a) -5 
(b) 10 
(c) -10 
(d) 5 
Ans: (b) lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? . We add and subtract ?? (?? )?? (?? ) in numerator 
 = lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )+?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]- lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]
 = ?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ]-?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ] = ?? (?? )?? '
(?? )-?? (?? )?? '
(?? )  [by using first principle formula] 
 = 3.4-(-1)(-2)= 12-2= 10
 
[by using first principle formula] 
Trick : lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? 
Using L-Hospital's rule, Limit =lim
?? ??? ?
?? '
(?? )?? (?? )-?? (?? )?? '
(?? )
1
; 
Limit =?? '
(?? )?? (?? )-?? (?? )?? '
(?? )=(4)(3)-(-1)(-2)=12-2=10. 
 
Q3:  If ?? ?? (?? )+?? ?? (
?? ?? )=?? +?? and ?? =???? (?? ) then (
????
????
)
?? =?? is equal to 
(a) 14 
(b) 
?? ?? 
(c) 1 
(d) None of these 
Ans: (b)  ?5?? (?? )+3?? (
1
?? )=?? +2   
Replacing ?? by 
1
?? in (i), 5?? (
1
?? )+3?? (?? )=
1
?? +2 
On solving equation (i) and (ii), we get, 16?? (?? )=5?? -
3
?? +4,?16?? '
(?? )=5+
3
?? 2
 
 ??? =???? (?? )?
????
????
=?? (?? )+?? ?? '
(?? )=
1
16
(5?? -
3
?? +4)+?? ·
1
16
(5+
3
?? 2
)
 at ?? =1,
????
????
=
1
16
(5-3+4)+
1
16
(5+3)=
7
8
.
 
Q4: The derivative of ?? (?? )=?? |
?? at ?? =?? is 
(a) ?? 
(b) 1 
(c) -1 
(d) Not defined 
Ans:   (a)  ?? (?? )={
?? 3
,?? =0
-?? 3
, ?? <0
  and  ?? '
(?? )={
3?? 2
,?? =0
-3?? 2
,?? <0
 
 
?? '
(0
+
)=?? '
(0
-
)=0 
 
Q5: Example: 6 The first derivative of the function (?????? ?? ?? ?????? ?? ?? ?????? ?? ?? +??????
?? ?? ?? +?? ) with 
respect to ?? at ?? =?? is 
(a) 2 
(b) -1 
(c) -?? +?? ?? ?????? ?? ?? 
(d) -?? +??????
?? ?? 
Ans: (b)  ?? (?? )=sin 2?? ·cos 2?? ·cos 3?? +log
2
 2
?? +3
, ?? (?? )=
1
2
sin 4?? cos 3?? +(?? +
3)log
2
 2, ?? (?? )=
1
4
[sin 7?? +sin ?? ]+?? +3 
Differentiate w.r.t. ?? , 
?? '
(?? )=
1
4
[7cos 7?? +cos ?? ]+1,?? '
(?? )=
1
4
7cos 7?? +
1
4
cos ?? +1, ?? '
(?? )=-2+1=-1. 
 
 
 
Q6: ?? is a point on the circumference of a circle & ?? is the foot of the perpendicular from ?? on a 
fixed diameter ???? . Then the limit of 
?? ?? ?? ????
 as ?? tends to ?? along the circumference 
(A) Does not exist 
(B) Equal to one 
(C) Is equal to the length ???? 
(D) None 
Ans: (C) 
Hint: ?? ?? 2
=???? ×????  
 
 Q8: ?????? ?? ??? ?(?? +??????
?????? 
?? ?? ?? ?????? ?? ) 
 
(A) Is equal to 4 (B) Is equal to 25 
(C) Is equal to 289 (D) Is non existent 
 
 
Ans: (C) 
 lim
?? ?0
?(1+log
cos 
?? 2
2
 cos ?? )
2
 lim
?? ?0
?log
cos 
?? 2
 cos ?? =lim
?? ?0
?
log (cos ?? )
log (cos 
?? 2
)
 =lim
?? ?0
?
-sin ?? cos ?? 1
2
sin ?? /2
cos ?? /2
=lim
?? ?0
?
4cos
2
 
?? 2
cos ?? =4
 
 
Q9: ?????? ?? ??? ?
(?? +?? )
?? ?? -?? (?? +?? )
?? ?? -?? is 
(A) 1 
(B) 0 
(C) ?? /?? 
(D) 8 
Ans: (C) Use lim
?? ??? ?
?? ?? -?? ?? ?? -?? =?? ?? ?? -1
 
Q10: Centre of circle is the limit of point of intersection of tines ?? ?? +?? ?? =?? and (?? +?? )?? +
?? ?? ?? ?? =?? as ?? tends to 1 . If it passes through (?? ,?? ) its radius is - 
(A) 
v????????
????
 
(B) 
????
????
 
(C) 
????????
v????
 
(D) v
????????
????
 
Ans: (A) On solving for x and y we get 
?? =lim
?? ?1
?
1-?? 2
2+?? -3?? 2
lim
?? ?1
?
(1-?? )(1+?? )
(2+3?? )(1-?? )
=
2
5
 ??? =-
1
25
 
Now, radius can be found by distance formula. 
 
Q11:  The function ?? (?? ) is defined as follows ?? (?? )={
?? if ?? <?? ?? ?? if ?? =?? =?? ?? ?? -?? +?? if ?? >?? then ?? (?? ) is 
(A) Derivable and cont. at ?? =?? 
(B) Derivable at ?? =?? but not continuous at ?? =?? 
(C) Neither derivable nor cont. at ?? =?? 
(D) Not derivable at ?? =?? but continuous at ?? =?? 
Ans: (D) ?? (?? )={
?? ?? <0
?? 2
, 0=?? =1
?? 3
-?? +1, ?? >0
 
At ?? =0 
LHL=lim
?? ?0
-??? (?? )=lim
?? ?0
-??? =0 
 
 
 
Q12:  A function ?? defined as ?? (?? )=?? [?? ] for -?? =?? =?? where [?? ] defines the greatest integer 
=?? is 
(A) Continuous at all points in the domain of ?? but nonderivable at a finite number of points 
(B) Discontinuous at all points & hence non -derivable at all points in the domain of ?? 
(C) Discontinuous at a finite number of points but not derivable at all points in the domain of ?? 
(D) Discontinuous & also non -derivable at a finite number of points of ?? . 
Ans: (D) Discontinuous & also non -derivable at a finite number of points of ?? . 
Q13: The value of ?????? ?? ??? ?
(?? +?? )
?? ?? -(?? -?? )
?? ?? ?? is: 
(A) ?? /?? 
(B) ?? /?? 
(C) 1 
(D) ?? /?? 
Ans: (A) 
Page 5


Solved Example on Limit Continuity and 
Derivative 
JEE Mains 
Q1:  If ?? (?? +?? )=?? (?? )·?? (?? ) for all ?? and ?? and ?? (?? )=?? ,?? '
(?? )=?? , then ?? '
(?? ) will be 
(a) 2 
(b) 4 
(c) 6 
(d) 8 
Ans: (c) Let ?? =5,?? =0??? (5+0)=?? (5)·?? (0) 
??? (5)=?? (5)?? (0)??? (0)=1 
Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] 
 Therefore, ?? '
(5)=lim
h?0
?
?? (5+h)-?? (5)
h
=lim
h?0
?
?? (5)?? (h)-?? (5)
h
=lim
h?0
?2[
?? (h)-1
h
] {??? (5)=2}
 =2lim
h?0
?[
?? (h)-?? (0)
h
]=2×?? '
(0)=2×3=6.
 
Q2:  If ?? (?? )=?? ,?? '
(?? )=-?? ,?? (?? )=-?? ,?? '
(?? )=?? , then ?????? ?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = 
(a) -5 
(b) 10 
(c) -10 
(d) 5 
Ans: (b) lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? . We add and subtract ?? (?? )?? (?? ) in numerator 
 = lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )+?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? = lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]- lim
?? ??? ??? (?? )[
?? (?? )-?? (?? )
?? -?? ]
 = ?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ]-?? (?? )lim
?? ??? ?[
?? (?? )-?? (?? )
?? -?? ] = ?? (?? )?? '
(?? )-?? (?? )?? '
(?? )  [by using first principle formula] 
 = 3.4-(-1)(-2)= 12-2= 10
 
[by using first principle formula] 
Trick : lim
?? ??? ?
?? (?? )?? (?? )-?? (?? )?? (?? )
?? -?? 
Using L-Hospital's rule, Limit =lim
?? ??? ?
?? '
(?? )?? (?? )-?? (?? )?? '
(?? )
1
; 
Limit =?? '
(?? )?? (?? )-?? (?? )?? '
(?? )=(4)(3)-(-1)(-2)=12-2=10. 
 
Q3:  If ?? ?? (?? )+?? ?? (
?? ?? )=?? +?? and ?? =???? (?? ) then (
????
????
)
?? =?? is equal to 
(a) 14 
(b) 
?? ?? 
(c) 1 
(d) None of these 
Ans: (b)  ?5?? (?? )+3?? (
1
?? )=?? +2   
Replacing ?? by 
1
?? in (i), 5?? (
1
?? )+3?? (?? )=
1
?? +2 
On solving equation (i) and (ii), we get, 16?? (?? )=5?? -
3
?? +4,?16?? '
(?? )=5+
3
?? 2
 
 ??? =???? (?? )?
????
????
=?? (?? )+?? ?? '
(?? )=
1
16
(5?? -
3
?? +4)+?? ·
1
16
(5+
3
?? 2
)
 at ?? =1,
????
????
=
1
16
(5-3+4)+
1
16
(5+3)=
7
8
.
 
Q4: The derivative of ?? (?? )=?? |
?? at ?? =?? is 
(a) ?? 
(b) 1 
(c) -1 
(d) Not defined 
Ans:   (a)  ?? (?? )={
?? 3
,?? =0
-?? 3
, ?? <0
  and  ?? '
(?? )={
3?? 2
,?? =0
-3?? 2
,?? <0
 
 
?? '
(0
+
)=?? '
(0
-
)=0 
 
Q5: Example: 6 The first derivative of the function (?????? ?? ?? ?????? ?? ?? ?????? ?? ?? +??????
?? ?? ?? +?? ) with 
respect to ?? at ?? =?? is 
(a) 2 
(b) -1 
(c) -?? +?? ?? ?????? ?? ?? 
(d) -?? +??????
?? ?? 
Ans: (b)  ?? (?? )=sin 2?? ·cos 2?? ·cos 3?? +log
2
 2
?? +3
, ?? (?? )=
1
2
sin 4?? cos 3?? +(?? +
3)log
2
 2, ?? (?? )=
1
4
[sin 7?? +sin ?? ]+?? +3 
Differentiate w.r.t. ?? , 
?? '
(?? )=
1
4
[7cos 7?? +cos ?? ]+1,?? '
(?? )=
1
4
7cos 7?? +
1
4
cos ?? +1, ?? '
(?? )=-2+1=-1. 
 
 
 
Q6: ?? is a point on the circumference of a circle & ?? is the foot of the perpendicular from ?? on a 
fixed diameter ???? . Then the limit of 
?? ?? ?? ????
 as ?? tends to ?? along the circumference 
(A) Does not exist 
(B) Equal to one 
(C) Is equal to the length ???? 
(D) None 
Ans: (C) 
Hint: ?? ?? 2
=???? ×????  
 
 Q8: ?????? ?? ??? ?(?? +??????
?????? 
?? ?? ?? ?????? ?? ) 
 
(A) Is equal to 4 (B) Is equal to 25 
(C) Is equal to 289 (D) Is non existent 
 
 
Ans: (C) 
 lim
?? ?0
?(1+log
cos 
?? 2
2
 cos ?? )
2
 lim
?? ?0
?log
cos 
?? 2
 cos ?? =lim
?? ?0
?
log (cos ?? )
log (cos 
?? 2
)
 =lim
?? ?0
?
-sin ?? cos ?? 1
2
sin ?? /2
cos ?? /2
=lim
?? ?0
?
4cos
2
 
?? 2
cos ?? =4
 
 
Q9: ?????? ?? ??? ?
(?? +?? )
?? ?? -?? (?? +?? )
?? ?? -?? is 
(A) 1 
(B) 0 
(C) ?? /?? 
(D) 8 
Ans: (C) Use lim
?? ??? ?
?? ?? -?? ?? ?? -?? =?? ?? ?? -1
 
Q10: Centre of circle is the limit of point of intersection of tines ?? ?? +?? ?? =?? and (?? +?? )?? +
?? ?? ?? ?? =?? as ?? tends to 1 . If it passes through (?? ,?? ) its radius is - 
(A) 
v????????
????
 
(B) 
????
????
 
(C) 
????????
v????
 
(D) v
????????
????
 
Ans: (A) On solving for x and y we get 
?? =lim
?? ?1
?
1-?? 2
2+?? -3?? 2
lim
?? ?1
?
(1-?? )(1+?? )
(2+3?? )(1-?? )
=
2
5
 ??? =-
1
25
 
Now, radius can be found by distance formula. 
 
Q11:  The function ?? (?? ) is defined as follows ?? (?? )={
?? if ?? <?? ?? ?? if ?? =?? =?? ?? ?? -?? +?? if ?? >?? then ?? (?? ) is 
(A) Derivable and cont. at ?? =?? 
(B) Derivable at ?? =?? but not continuous at ?? =?? 
(C) Neither derivable nor cont. at ?? =?? 
(D) Not derivable at ?? =?? but continuous at ?? =?? 
Ans: (D) ?? (?? )={
?? ?? <0
?? 2
, 0=?? =1
?? 3
-?? +1, ?? >0
 
At ?? =0 
LHL=lim
?? ?0
-??? (?? )=lim
?? ?0
-??? =0 
 
 
 
Q12:  A function ?? defined as ?? (?? )=?? [?? ] for -?? =?? =?? where [?? ] defines the greatest integer 
=?? is 
(A) Continuous at all points in the domain of ?? but nonderivable at a finite number of points 
(B) Discontinuous at all points & hence non -derivable at all points in the domain of ?? 
(C) Discontinuous at a finite number of points but not derivable at all points in the domain of ?? 
(D) Discontinuous & also non -derivable at a finite number of points of ?? . 
Ans: (D) Discontinuous & also non -derivable at a finite number of points of ?? . 
Q13: The value of ?????? ?? ??? ?
(?? +?? )
?? ?? -(?? -?? )
?? ?? ?? is: 
(A) ?? /?? 
(B) ?? /?? 
(C) 1 
(D) ?? /?? 
Ans: (A) 
 lim
?? ?0
?
(1+?? )
1
3
-(1-?? )
1
3
?? (1+?? )
2/3
+(1-?? )
2/3
+(1+?? )
1/3
(1-?? )
1/3
(1+?? )
2/3
+(1-?? )
2/3
+(1+?? )
1/3
(1-?? )
1/3
 =lim
?? ?0
?
(1+?? )-(1-?? )
(1+?? )
2/3
+(1-?? )
2/3
+(1+?? )
1/3
(1-?? )
1/3
·
1
?? =lim
?? ?0
?
2
(1+?? )
2/3
+(1-?? )
2/3
+(1+?? )
1/3
(1-?? )
1/3
=
2
1+1+1
=
2
3
 
Hence, (A) is the correct answer. 
 
 
Q14: The values of ?? and ?? so that function ?? (?? ) defined by ?? (?? )=
{
 
 
?? +?? v?? ?????? ?? , ?? =?? <
?? ?? ?? ?? ?????? ?? +?? ,
?? ?? =?? <
?? ?? ?? ?????? ?? ?? -?? ?????? ?? ,
?? ?? =?? =?? become continuous, respectively are 
(A) 
-?? ????
,
?? ?? 
(B) 
?? ?? ,
-?? ????
 
(C) 
?? ????
,
?? ?? 
(D) 
?? ?? ,
?? ????
 
Ans: (B) 
?? (?? )={
?? +?? v2sin ?? 0=?? <?? /4
2?? cot ?? +?? ?? /4=?? <?? /2
?? cos 2?? -?? sin ?? ?? /2=?? =?? 
L.H. limit at ?? <
?? 4
 
 = lim
?? ??? /4
?x+Av2sin x=
?? 4
+Av2sin 
?? 4
=
?? 4
+Av2×
1
v2
=A+
?? 4
 R.H. limit = lim
x??? /4
?2xcot x+B=
2?? 4
·cot 
?? 4
+B=
?? 2
+B
 
?? +
?? 4
=?? +
?? 2
??? -?? =
?? 4
 
 L.H. limit at ?? <
?? 2
 = lim
?? ??? /2
?-(2?? cot ?? +?? )=2×
?? 2
cot 
?? 2
+?? =?? RH limit =?? lim
?? ??? /2
?+?? cos 2?? -?? sin ?? =?? cos ?? -?? sin 
?? 2
 
Q15: If ?? (?? )=(
?? ?? +?? ?? +?? ?? ?? +?? +?? )
?? , then ?????? ?? ?8
??? (?? ) is 
(A) ?? ?? 
(B) ?? ??