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JEE Main Previous Year Questions (2025): Limits and Derivatives
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Page 1
JEE Main Previous Year Questions
(2025): Limits and Derivatives
Q1: If the function ?? ( ?? ) =
?????? ? ( ?????? ? ?? ) - ?? ?? ?? ? ( ?? ?? ?? ? ?? )
?????? ? ?? - ?? ?? ?? ? ?? is continuous at ?? = ?? , then ?? ( ?? ) is equal
to _ _ _ _
Ans: B
li m
?? ? 0
?
t a n ? ( t a n ? ?? ) - t a n ? ?? t a n
3
? ?? t a n
3
? ?? ?? 3
+
t a n ? ?? - s i n ? ?? ?? 3
+
s i n ? ?? - s i n ? ( s i n ? ?? )
s i n
3
? ?? s i n
3
? ?? ?? 3
t a n ? ?? - s i n ? ?? ?? 3
=
1
3
+
1
2
+
1
6
1
2
= 2
Q2: If ?????? ?? ? ?? ?
?? ?? ?? ? ( ?? ?? ) + ?? ?? ?? ?? ? ( ?? ?? ) - ?? ?? ?? is finite, then ( ?? + ?? ) is equal to :
A.
1
2
B. 0
C.
3
4
D. -1
Ans: A
li m
?? ? 0
?
c os ? 2 ?? + ?? c os ? 4 ?? - ?? ?? 4
= finite
?? =
{ 1 -
( 2 ?? )
2
2
+
( 2 ?? )
4
4 ? } } + ?? { 1 -
( 4 ?? )
2
2
+
( 4 ?? )
4
4
? } - ?? ?? 4
?? =
( 1 + ?? - ?? ) - ?? 2
( 2 + 8 ?? ) + ?? 4
(
2
3
+
32
3
?? ) + ?? 6
( ) …
?? 4
? 1 + a - b = 0 and 2 + 8a = 0 ? a = -
1
4
b = a + 1
= -
1
4
+ 1 =
3
4
? a + b = -
1
4
+
3
4
=
1
2
Q3: ?????? ?? ? ?? + ?
?????? ? ( ?? ( ?? )
?? ?? ) ???? ?? ?? ? ( ?? + ?? ?? ?? )
( ??????
- ?? ? ?? v ?? )
?? ( ?? ?? ( ?? )
?? ?? - ?? )
is equal to
A.
1
15
Page 2
JEE Main Previous Year Questions
(2025): Limits and Derivatives
Q1: If the function ?? ( ?? ) =
?????? ? ( ?????? ? ?? ) - ?? ?? ?? ? ( ?? ?? ?? ? ?? )
?????? ? ?? - ?? ?? ?? ? ?? is continuous at ?? = ?? , then ?? ( ?? ) is equal
to _ _ _ _
Ans: B
li m
?? ? 0
?
t a n ? ( t a n ? ?? ) - t a n ? ?? t a n
3
? ?? t a n
3
? ?? ?? 3
+
t a n ? ?? - s i n ? ?? ?? 3
+
s i n ? ?? - s i n ? ( s i n ? ?? )
s i n
3
? ?? s i n
3
? ?? ?? 3
t a n ? ?? - s i n ? ?? ?? 3
=
1
3
+
1
2
+
1
6
1
2
= 2
Q2: If ?????? ?? ? ?? ?
?? ?? ?? ? ( ?? ?? ) + ?? ?? ?? ?? ? ( ?? ?? ) - ?? ?? ?? is finite, then ( ?? + ?? ) is equal to :
A.
1
2
B. 0
C.
3
4
D. -1
Ans: A
li m
?? ? 0
?
c os ? 2 ?? + ?? c os ? 4 ?? - ?? ?? 4
= finite
?? =
{ 1 -
( 2 ?? )
2
2
+
( 2 ?? )
4
4 ? } } + ?? { 1 -
( 4 ?? )
2
2
+
( 4 ?? )
4
4
? } - ?? ?? 4
?? =
( 1 + ?? - ?? ) - ?? 2
( 2 + 8 ?? ) + ?? 4
(
2
3
+
32
3
?? ) + ?? 6
( ) …
?? 4
? 1 + a - b = 0 and 2 + 8a = 0 ? a = -
1
4
b = a + 1
= -
1
4
+ 1 =
3
4
? a + b = -
1
4
+
3
4
=
1
2
Q3: ?????? ?? ? ?? + ?
?????? ? ( ?? ( ?? )
?? ?? ) ???? ?? ?? ? ( ?? + ?? ?? ?? )
( ??????
- ?? ? ?? v ?? )
?? ( ?? ?? ( ?? )
?? ?? - ?? )
is equal to
A.
1
15
B. 1
C.
1
3
D.
5
3
Ans: C
li m
?? ? 0
? (
tan ? ( 5 x
1 / 3
)
5 x
1 / 3
) · (
( 3 v x )
2
( tan
- 1
? 3 v x )
2
) (
l ( 1 + 3 x
2
)
3 x
2
) (
5 x
4 / 3
e
5 ?? 4
- 1
) ×
5 x
1 / 3
· 3 x
2
5 x
4 / 3
· 9x
=
1
3
Q4: For ?? , ?? , ?? , ? ?? , if ?????? ?? ? ?? ?
?? ?? ?? ?? ?? ? ???? + ( ?? - ?? ) ?? ?? ?? ?? ?? ?? ? ?? ?? - ????
= ?? , then ?? + ?? - ?? is equal to:
A. 7
B. 4
C. 6
D. -1
Ans: A
li m
?? ? 10
?
?? 2
( ???? ) + ( ?? - 1 ) ( 1 +
?? 2
1
)
2 ?? -
8 ?? 3
6
- ????
= 3
li m
?? ? 0
?
( ?? - 1 ) + ( ?? - 1 ) ?? 2
+ ?? ?? 3
( 2 - ?? ) ?? -
4
3
?? 3
= 3
?? - 1 , ?? = 2 ,
- 3 ?? 4
= + 3 ? ?? = - 4
?? + ?? - ?? = 7
Q5. Given below are two statements :
Statement I : ?????? ?? ? ?? ? (
??????
- ?? ? ?? + ???? ?? ?? ? v
?? + ?? ?? - ?? - ?? ?? ?? ?? ) =
?? ??
Statement II : ?????? ?? ? ?? ? ( ?? ?? ?? - ?? ) =
?? ?? ??
In the light of the above statements, choose the correct answer from the options
given below :
A. Statement I is false but Statement II is true
B. Statement I is true but Statement II is false
C. Both Statement I and Statement II are false
D. Both Statement I and Statement II are true
Ans: D
Page 3
JEE Main Previous Year Questions
(2025): Limits and Derivatives
Q1: If the function ?? ( ?? ) =
?????? ? ( ?????? ? ?? ) - ?? ?? ?? ? ( ?? ?? ?? ? ?? )
?????? ? ?? - ?? ?? ?? ? ?? is continuous at ?? = ?? , then ?? ( ?? ) is equal
to _ _ _ _
Ans: B
li m
?? ? 0
?
t a n ? ( t a n ? ?? ) - t a n ? ?? t a n
3
? ?? t a n
3
? ?? ?? 3
+
t a n ? ?? - s i n ? ?? ?? 3
+
s i n ? ?? - s i n ? ( s i n ? ?? )
s i n
3
? ?? s i n
3
? ?? ?? 3
t a n ? ?? - s i n ? ?? ?? 3
=
1
3
+
1
2
+
1
6
1
2
= 2
Q2: If ?????? ?? ? ?? ?
?? ?? ?? ? ( ?? ?? ) + ?? ?? ?? ?? ? ( ?? ?? ) - ?? ?? ?? is finite, then ( ?? + ?? ) is equal to :
A.
1
2
B. 0
C.
3
4
D. -1
Ans: A
li m
?? ? 0
?
c os ? 2 ?? + ?? c os ? 4 ?? - ?? ?? 4
= finite
?? =
{ 1 -
( 2 ?? )
2
2
+
( 2 ?? )
4
4 ? } } + ?? { 1 -
( 4 ?? )
2
2
+
( 4 ?? )
4
4
? } - ?? ?? 4
?? =
( 1 + ?? - ?? ) - ?? 2
( 2 + 8 ?? ) + ?? 4
(
2
3
+
32
3
?? ) + ?? 6
( ) …
?? 4
? 1 + a - b = 0 and 2 + 8a = 0 ? a = -
1
4
b = a + 1
= -
1
4
+ 1 =
3
4
? a + b = -
1
4
+
3
4
=
1
2
Q3: ?????? ?? ? ?? + ?
?????? ? ( ?? ( ?? )
?? ?? ) ???? ?? ?? ? ( ?? + ?? ?? ?? )
( ??????
- ?? ? ?? v ?? )
?? ( ?? ?? ( ?? )
?? ?? - ?? )
is equal to
A.
1
15
B. 1
C.
1
3
D.
5
3
Ans: C
li m
?? ? 0
? (
tan ? ( 5 x
1 / 3
)
5 x
1 / 3
) · (
( 3 v x )
2
( tan
- 1
? 3 v x )
2
) (
l ( 1 + 3 x
2
)
3 x
2
) (
5 x
4 / 3
e
5 ?? 4
- 1
) ×
5 x
1 / 3
· 3 x
2
5 x
4 / 3
· 9x
=
1
3
Q4: For ?? , ?? , ?? , ? ?? , if ?????? ?? ? ?? ?
?? ?? ?? ?? ?? ? ???? + ( ?? - ?? ) ?? ?? ?? ?? ?? ?? ? ?? ?? - ????
= ?? , then ?? + ?? - ?? is equal to:
A. 7
B. 4
C. 6
D. -1
Ans: A
li m
?? ? 10
?
?? 2
( ???? ) + ( ?? - 1 ) ( 1 +
?? 2
1
)
2 ?? -
8 ?? 3
6
- ????
= 3
li m
?? ? 0
?
( ?? - 1 ) + ( ?? - 1 ) ?? 2
+ ?? ?? 3
( 2 - ?? ) ?? -
4
3
?? 3
= 3
?? - 1 , ?? = 2 ,
- 3 ?? 4
= + 3 ? ?? = - 4
?? + ?? - ?? = 7
Q5. Given below are two statements :
Statement I : ?????? ?? ? ?? ? (
??????
- ?? ? ?? + ???? ?? ?? ? v
?? + ?? ?? - ?? - ?? ?? ?? ?? ) =
?? ??
Statement II : ?????? ?? ? ?? ? ( ?? ?? ?? - ?? ) =
?? ?? ??
In the light of the above statements, choose the correct answer from the options
given below :
A. Statement I is false but Statement II is true
B. Statement I is true but Statement II is false
C. Both Statement I and Statement II are false
D. Both Statement I and Statement II are true
Ans: D
li m
?? ? 0
?
tan
- 1
? ?? +
1
2
[ ln ? ( 1 + ?? ) - ln ? ( 1 - ?? ) ] - 2 ?? ?? 5
= li m
?? ? 0
?
( ?? -
?? 3
3
+
?? 5
5
… ) +
1
2
[ ?? -
?? 2
2
+
?? 3
3
… - ( - ?? -
?? 2
2
-
?? 3
3
… ) ] - 2 ?? ?? 5
= li m
?? ? 0
?
2 ?? +
2 ?? 5
5
? - 2 ?? ?? 5
=
2
5
li m
?? ? 1
? ?? 2
( 1 - ?? )
= ?? l i m
?? ? ?? ? (
2
( 1 - ?? )
) ( ?? - 1 )
= ?? - 2
? Both statements correct
Q6: If ?? ???? ?? ? ?? (
?????? ? ?? ?? )
?? ?? ?? = ?? , then ???? ???? ?? ?? ? ?? is equal to _ _ _ _
Ans: 32
?? = li m
?? ? 0
? (
tan ? ?? ?? )
1
?? 2
? P = e
l i m
x ? 0
? (
tan ? ?? - x
x
3
)
= e
l i m
?? ? 0
?
( ?? +
?? 3
3
+
2 ?? 5
15
+ ? . . - ?? )
?? 3
= ?? 1 / 3
? 96 log
e
p
= 96 ×
1
3
= 32
Q7: If ?????? ?? ? ?? + ?
( ?? - ?? ) ( ?? + ?? ?? ?? ?? ? ( ?? - ?? ) ) + ?? ?? ?? ?? ? ( ?? - ?? )
( ?? - ?? )
?? = - ?? , where ?? , ?? ? R, then ?? + ?? is equal
to
A. 18
B. 20
C. 19
D. 17
Ans: A
Put ?? = 1 + h
li m
h ? 0
?
h ( 6 + ?? c osh ) - ?? s i n h
h
3
= - 1
li m
h ? 0
?
h ( 6 + ?? ( 1 -
h
2
2 !
) ) - ?? ( h -
h
3
3 !
)
h
3
= - 1
6 + ?? - ?? = 0 and -
?? 2
+
?? 6
= - 1
?? + ?? = 18
Page 4
JEE Main Previous Year Questions
(2025): Limits and Derivatives
Q1: If the function ?? ( ?? ) =
?????? ? ( ?????? ? ?? ) - ?? ?? ?? ? ( ?? ?? ?? ? ?? )
?????? ? ?? - ?? ?? ?? ? ?? is continuous at ?? = ?? , then ?? ( ?? ) is equal
to _ _ _ _
Ans: B
li m
?? ? 0
?
t a n ? ( t a n ? ?? ) - t a n ? ?? t a n
3
? ?? t a n
3
? ?? ?? 3
+
t a n ? ?? - s i n ? ?? ?? 3
+
s i n ? ?? - s i n ? ( s i n ? ?? )
s i n
3
? ?? s i n
3
? ?? ?? 3
t a n ? ?? - s i n ? ?? ?? 3
=
1
3
+
1
2
+
1
6
1
2
= 2
Q2: If ?????? ?? ? ?? ?
?? ?? ?? ? ( ?? ?? ) + ?? ?? ?? ?? ? ( ?? ?? ) - ?? ?? ?? is finite, then ( ?? + ?? ) is equal to :
A.
1
2
B. 0
C.
3
4
D. -1
Ans: A
li m
?? ? 0
?
c os ? 2 ?? + ?? c os ? 4 ?? - ?? ?? 4
= finite
?? =
{ 1 -
( 2 ?? )
2
2
+
( 2 ?? )
4
4 ? } } + ?? { 1 -
( 4 ?? )
2
2
+
( 4 ?? )
4
4
? } - ?? ?? 4
?? =
( 1 + ?? - ?? ) - ?? 2
( 2 + 8 ?? ) + ?? 4
(
2
3
+
32
3
?? ) + ?? 6
( ) …
?? 4
? 1 + a - b = 0 and 2 + 8a = 0 ? a = -
1
4
b = a + 1
= -
1
4
+ 1 =
3
4
? a + b = -
1
4
+
3
4
=
1
2
Q3: ?????? ?? ? ?? + ?
?????? ? ( ?? ( ?? )
?? ?? ) ???? ?? ?? ? ( ?? + ?? ?? ?? )
( ??????
- ?? ? ?? v ?? )
?? ( ?? ?? ( ?? )
?? ?? - ?? )
is equal to
A.
1
15
B. 1
C.
1
3
D.
5
3
Ans: C
li m
?? ? 0
? (
tan ? ( 5 x
1 / 3
)
5 x
1 / 3
) · (
( 3 v x )
2
( tan
- 1
? 3 v x )
2
) (
l ( 1 + 3 x
2
)
3 x
2
) (
5 x
4 / 3
e
5 ?? 4
- 1
) ×
5 x
1 / 3
· 3 x
2
5 x
4 / 3
· 9x
=
1
3
Q4: For ?? , ?? , ?? , ? ?? , if ?????? ?? ? ?? ?
?? ?? ?? ?? ?? ? ???? + ( ?? - ?? ) ?? ?? ?? ?? ?? ?? ? ?? ?? - ????
= ?? , then ?? + ?? - ?? is equal to:
A. 7
B. 4
C. 6
D. -1
Ans: A
li m
?? ? 10
?
?? 2
( ???? ) + ( ?? - 1 ) ( 1 +
?? 2
1
)
2 ?? -
8 ?? 3
6
- ????
= 3
li m
?? ? 0
?
( ?? - 1 ) + ( ?? - 1 ) ?? 2
+ ?? ?? 3
( 2 - ?? ) ?? -
4
3
?? 3
= 3
?? - 1 , ?? = 2 ,
- 3 ?? 4
= + 3 ? ?? = - 4
?? + ?? - ?? = 7
Q5. Given below are two statements :
Statement I : ?????? ?? ? ?? ? (
??????
- ?? ? ?? + ???? ?? ?? ? v
?? + ?? ?? - ?? - ?? ?? ?? ?? ) =
?? ??
Statement II : ?????? ?? ? ?? ? ( ?? ?? ?? - ?? ) =
?? ?? ??
In the light of the above statements, choose the correct answer from the options
given below :
A. Statement I is false but Statement II is true
B. Statement I is true but Statement II is false
C. Both Statement I and Statement II are false
D. Both Statement I and Statement II are true
Ans: D
li m
?? ? 0
?
tan
- 1
? ?? +
1
2
[ ln ? ( 1 + ?? ) - ln ? ( 1 - ?? ) ] - 2 ?? ?? 5
= li m
?? ? 0
?
( ?? -
?? 3
3
+
?? 5
5
… ) +
1
2
[ ?? -
?? 2
2
+
?? 3
3
… - ( - ?? -
?? 2
2
-
?? 3
3
… ) ] - 2 ?? ?? 5
= li m
?? ? 0
?
2 ?? +
2 ?? 5
5
? - 2 ?? ?? 5
=
2
5
li m
?? ? 1
? ?? 2
( 1 - ?? )
= ?? l i m
?? ? ?? ? (
2
( 1 - ?? )
) ( ?? - 1 )
= ?? - 2
? Both statements correct
Q6: If ?? ???? ?? ? ?? (
?????? ? ?? ?? )
?? ?? ?? = ?? , then ???? ???? ?? ?? ? ?? is equal to _ _ _ _
Ans: 32
?? = li m
?? ? 0
? (
tan ? ?? ?? )
1
?? 2
? P = e
l i m
x ? 0
? (
tan ? ?? - x
x
3
)
= e
l i m
?? ? 0
?
( ?? +
?? 3
3
+
2 ?? 5
15
+ ? . . - ?? )
?? 3
= ?? 1 / 3
? 96 log
e
p
= 96 ×
1
3
= 32
Q7: If ?????? ?? ? ?? + ?
( ?? - ?? ) ( ?? + ?? ?? ?? ?? ? ( ?? - ?? ) ) + ?? ?? ?? ?? ? ( ?? - ?? )
( ?? - ?? )
?? = - ?? , where ?? , ?? ? R, then ?? + ?? is equal
to
A. 18
B. 20
C. 19
D. 17
Ans: A
Put ?? = 1 + h
li m
h ? 0
?
h ( 6 + ?? c osh ) - ?? s i n h
h
3
= - 1
li m
h ? 0
?
h ( 6 + ?? ( 1 -
h
2
2 !
) ) - ?? ( h -
h
3
3 !
)
h
3
= - 1
6 + ?? - ?? = 0 and -
?? 2
+
?? 6
= - 1
?? + ?? = 18
Q8: Let ?? and ?? be the number of points at which the function ?? ( ?? ) =
?? ???? { ?? , ?? ?? , ?? ?? , … , ?? ????
} , ?? ? R, is not differentiable and not continuous, respectively.
Then ?? + ?? is equal to _ _ _ _ .
Ans: 3
Solution:
?? ( ?? ) = {
?? , ?? < - 1
?? 21
, - 1 = ?? < 0
?? , 0 = ?? < 1
?? 21
, ?? = 1
?? ( ?? ) is continuous everywhere.
? n = 0
f
'
( x ) = {
1 , x < - 1
21 x
20
, - 1 = x < 0
1 , 0 < x < 1
21 x
20
, x = 1
? f ( x ) is non-differentiable at x = - 1 , 0 , 1
? m = 3
m + n = 3
Q9: Let ?? ( ?? ) = {
( ?? + ???? )
?? / ?? , ?? < ?? ?? + ?? , ?? = ?? ( ?? + ?? )
?? / ?? - ?? ( ?? + ?? )
?? / ?? - ?? ,
be continuous at ?? = ?? . Then ?? ?? ???? is equal to
A. 64
B. 72
C. 48
D. 36
Ans: C
?? ( 0
-
) = e
l i m
?? ? 0
?
ax
x
= e
a
f ( 0 ) = 1 + b
f ( 0
+
) =
1
2 v x + 4
1
3
( x + c )
-
2
3
=
1
2 ( 2 )
1
3
· c
-
2
3
=
3
4
?? 2 / 3
Also at x = 0;
?? 1 / 3
= 2 ? ?? = 8
So f ( 0
+
) =
3
4
( 8 )
2 / 3
= 3
Now, e
a
= b + 1 = 3
e
a
· b · c = 3 · 2 · 8 = 4 8
Page 5
JEE Main Previous Year Questions
(2025): Limits and Derivatives
Q1: If the function ?? ( ?? ) =
?????? ? ( ?????? ? ?? ) - ?? ?? ?? ? ( ?? ?? ?? ? ?? )
?????? ? ?? - ?? ?? ?? ? ?? is continuous at ?? = ?? , then ?? ( ?? ) is equal
to _ _ _ _
Ans: B
li m
?? ? 0
?
t a n ? ( t a n ? ?? ) - t a n ? ?? t a n
3
? ?? t a n
3
? ?? ?? 3
+
t a n ? ?? - s i n ? ?? ?? 3
+
s i n ? ?? - s i n ? ( s i n ? ?? )
s i n
3
? ?? s i n
3
? ?? ?? 3
t a n ? ?? - s i n ? ?? ?? 3
=
1
3
+
1
2
+
1
6
1
2
= 2
Q2: If ?????? ?? ? ?? ?
?? ?? ?? ? ( ?? ?? ) + ?? ?? ?? ?? ? ( ?? ?? ) - ?? ?? ?? is finite, then ( ?? + ?? ) is equal to :
A.
1
2
B. 0
C.
3
4
D. -1
Ans: A
li m
?? ? 0
?
c os ? 2 ?? + ?? c os ? 4 ?? - ?? ?? 4
= finite
?? =
{ 1 -
( 2 ?? )
2
2
+
( 2 ?? )
4
4 ? } } + ?? { 1 -
( 4 ?? )
2
2
+
( 4 ?? )
4
4
? } - ?? ?? 4
?? =
( 1 + ?? - ?? ) - ?? 2
( 2 + 8 ?? ) + ?? 4
(
2
3
+
32
3
?? ) + ?? 6
( ) …
?? 4
? 1 + a - b = 0 and 2 + 8a = 0 ? a = -
1
4
b = a + 1
= -
1
4
+ 1 =
3
4
? a + b = -
1
4
+
3
4
=
1
2
Q3: ?????? ?? ? ?? + ?
?????? ? ( ?? ( ?? )
?? ?? ) ???? ?? ?? ? ( ?? + ?? ?? ?? )
( ??????
- ?? ? ?? v ?? )
?? ( ?? ?? ( ?? )
?? ?? - ?? )
is equal to
A.
1
15
B. 1
C.
1
3
D.
5
3
Ans: C
li m
?? ? 0
? (
tan ? ( 5 x
1 / 3
)
5 x
1 / 3
) · (
( 3 v x )
2
( tan
- 1
? 3 v x )
2
) (
l ( 1 + 3 x
2
)
3 x
2
) (
5 x
4 / 3
e
5 ?? 4
- 1
) ×
5 x
1 / 3
· 3 x
2
5 x
4 / 3
· 9x
=
1
3
Q4: For ?? , ?? , ?? , ? ?? , if ?????? ?? ? ?? ?
?? ?? ?? ?? ?? ? ???? + ( ?? - ?? ) ?? ?? ?? ?? ?? ?? ? ?? ?? - ????
= ?? , then ?? + ?? - ?? is equal to:
A. 7
B. 4
C. 6
D. -1
Ans: A
li m
?? ? 10
?
?? 2
( ???? ) + ( ?? - 1 ) ( 1 +
?? 2
1
)
2 ?? -
8 ?? 3
6
- ????
= 3
li m
?? ? 0
?
( ?? - 1 ) + ( ?? - 1 ) ?? 2
+ ?? ?? 3
( 2 - ?? ) ?? -
4
3
?? 3
= 3
?? - 1 , ?? = 2 ,
- 3 ?? 4
= + 3 ? ?? = - 4
?? + ?? - ?? = 7
Q5. Given below are two statements :
Statement I : ?????? ?? ? ?? ? (
??????
- ?? ? ?? + ???? ?? ?? ? v
?? + ?? ?? - ?? - ?? ?? ?? ?? ) =
?? ??
Statement II : ?????? ?? ? ?? ? ( ?? ?? ?? - ?? ) =
?? ?? ??
In the light of the above statements, choose the correct answer from the options
given below :
A. Statement I is false but Statement II is true
B. Statement I is true but Statement II is false
C. Both Statement I and Statement II are false
D. Both Statement I and Statement II are true
Ans: D
li m
?? ? 0
?
tan
- 1
? ?? +
1
2
[ ln ? ( 1 + ?? ) - ln ? ( 1 - ?? ) ] - 2 ?? ?? 5
= li m
?? ? 0
?
( ?? -
?? 3
3
+
?? 5
5
… ) +
1
2
[ ?? -
?? 2
2
+
?? 3
3
… - ( - ?? -
?? 2
2
-
?? 3
3
… ) ] - 2 ?? ?? 5
= li m
?? ? 0
?
2 ?? +
2 ?? 5
5
? - 2 ?? ?? 5
=
2
5
li m
?? ? 1
? ?? 2
( 1 - ?? )
= ?? l i m
?? ? ?? ? (
2
( 1 - ?? )
) ( ?? - 1 )
= ?? - 2
? Both statements correct
Q6: If ?? ???? ?? ? ?? (
?????? ? ?? ?? )
?? ?? ?? = ?? , then ???? ???? ?? ?? ? ?? is equal to _ _ _ _
Ans: 32
?? = li m
?? ? 0
? (
tan ? ?? ?? )
1
?? 2
? P = e
l i m
x ? 0
? (
tan ? ?? - x
x
3
)
= e
l i m
?? ? 0
?
( ?? +
?? 3
3
+
2 ?? 5
15
+ ? . . - ?? )
?? 3
= ?? 1 / 3
? 96 log
e
p
= 96 ×
1
3
= 32
Q7: If ?????? ?? ? ?? + ?
( ?? - ?? ) ( ?? + ?? ?? ?? ?? ? ( ?? - ?? ) ) + ?? ?? ?? ?? ? ( ?? - ?? )
( ?? - ?? )
?? = - ?? , where ?? , ?? ? R, then ?? + ?? is equal
to
A. 18
B. 20
C. 19
D. 17
Ans: A
Put ?? = 1 + h
li m
h ? 0
?
h ( 6 + ?? c osh ) - ?? s i n h
h
3
= - 1
li m
h ? 0
?
h ( 6 + ?? ( 1 -
h
2
2 !
) ) - ?? ( h -
h
3
3 !
)
h
3
= - 1
6 + ?? - ?? = 0 and -
?? 2
+
?? 6
= - 1
?? + ?? = 18
Q8: Let ?? and ?? be the number of points at which the function ?? ( ?? ) =
?? ???? { ?? , ?? ?? , ?? ?? , … , ?? ????
} , ?? ? R, is not differentiable and not continuous, respectively.
Then ?? + ?? is equal to _ _ _ _ .
Ans: 3
Solution:
?? ( ?? ) = {
?? , ?? < - 1
?? 21
, - 1 = ?? < 0
?? , 0 = ?? < 1
?? 21
, ?? = 1
?? ( ?? ) is continuous everywhere.
? n = 0
f
'
( x ) = {
1 , x < - 1
21 x
20
, - 1 = x < 0
1 , 0 < x < 1
21 x
20
, x = 1
? f ( x ) is non-differentiable at x = - 1 , 0 , 1
? m = 3
m + n = 3
Q9: Let ?? ( ?? ) = {
( ?? + ???? )
?? / ?? , ?? < ?? ?? + ?? , ?? = ?? ( ?? + ?? )
?? / ?? - ?? ( ?? + ?? )
?? / ?? - ?? ,
be continuous at ?? = ?? . Then ?? ?? ???? is equal to
A. 64
B. 72
C. 48
D. 36
Ans: C
?? ( 0
-
) = e
l i m
?? ? 0
?
ax
x
= e
a
f ( 0 ) = 1 + b
f ( 0
+
) =
1
2 v x + 4
1
3
( x + c )
-
2
3
=
1
2 ( 2 )
1
3
· c
-
2
3
=
3
4
?? 2 / 3
Also at x = 0;
?? 1 / 3
= 2 ? ?? = 8
So f ( 0
+
) =
3
4
( 8 )
2 / 3
= 3
Now, e
a
= b + 1 = 3
e
a
· b · c = 3 · 2 · 8 = 4 8
Q10: The number of points of discontinuity of the function ?? ( ?? ) = [
?? ?? ?? ] - [ v ?? ] , ?? ?
[ ?? , ?? ], where [ · ] denotes the greatest integer function is _ _ _ _
Ans: 8
Solution:
Check for [
?? 2
2
] and [ v ?? ] becomes integers.
{ 0 , 1 , v 2 , 2 , v 6 , v 8 , v 10 , v 12 , v 14 , 4 }
Continuous at 0
+
, continuous at 4
-
[
?? 2
2
] = [ v ?? ], occurs at ?? = v 2
? Not continuous
Read MoreFAQs on JEE Main Previous Year Questions (2025): Limits and Derivatives
| 1. What are limits in calculus, and why are they important in understanding derivatives? |  |
Ans. Limits are a fundamental concept in calculus that describe the behavior of a function as it approaches a particular point or value. They are crucial for defining derivatives, which represent the rate of change of a function. Understanding limits allows students to analyze how functions behave near points of interest, especially where they might not be defined, thus providing a foundation for further studies in calculus, such as continuity and differentiability.
| 2. How do you find the derivative of a function using the limit definition? | |
Ans. The derivative of a function f(x) at a point x = a can be found using the limit definition: f'(a) = lim(h→0) [(f(a + h) - f(a)) / h]. This expression calculates the slope of the tangent line to the curve at the point x = a by considering the average rate of change over a small interval around a. If this limit exists, it gives the instantaneous rate of change at that point.
| 3. What are some common rules for finding derivatives that can simplify calculations? | |
Ans. There are several basic rules for finding derivatives that can simplify calculations: 1. Power Rule: If f(x) = xⁿ, then f'(x) = n*x^(n-1). 2. Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). 3. Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². 4. Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x). These rules help streamline the process of differentiation for various types of functions.
| 4. What is the significance of continuity in relation to limits and derivatives? | |
Ans. Continuity is significant because a function must be continuous at a point for a derivative to exist at that point. A function f(x) is continuous at x = a if lim(x→a) f(x) = f(a). If a function has a discontinuity at a point, the limit might not equal the function's value there, which implies that the derivative cannot be defined. Thus, continuity ensures that the behavior of the function is predictable in the vicinity of that point.
| 5. Can you explain the concept of one-sided limits and how they relate to derivatives? | |
Ans. One-sided limits are limits that consider the behavior of a function as it approaches a certain point from one side only: the left-hand limit (approaching from the left) and the right-hand limit (approaching from the right). Mathematically, they are represented as lim(x→a⁻) f(x) and lim(x→a⁺) f(x). For a derivative to be defined at a point x = a, both one-sided limits must exist and be equal. If they are not equal, the derivative at that point is undefined, indicating a potential cusp or vertical tangent.
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