Limits and Derivatives: JEE Mains Previous Year Questions (2021-2024)

# Limits and Derivatives: JEE Mains Previous Year Questions (2021-2024) | Mathematics (Maths) for JEE Main & Advanced PDF Download

``` Page 1

JEE Mains Previous Year Questions
(2021-2024): Limits and Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
Let { ?? } denote the fractional part of x and ?? ( ?? ) =
c o s
- 1
? ( 1 - { ?? }
2
) s i n
- 1
? ( 1 - { ?? } )
{ ?? } - { ?? }
3
, ?? ? 0. If ?? and R
respectively denotes the left hand limit and the right hand limit of ?? ( ?? ) at ?? = 0, then
32
?? 2
( ?? 2
+ ?? 2
) is equal to
Q2 - 2024 (01 Feb Shift 2)
Let ?? ( ?? ) = {
?? - 1 , ?? is even,
2 ?? , ?? is odd,
?? . If for some ?? ? ?? , ?? ( ?? ( ?? ( ?? ) ) ) = 21, then l i m
?? ? ?? - ? {
| ?? |
3
?? -
[
?? ?? ] }, where [ ?? ] denotes the greatest integer less than or equal to ?? , is equal to :
(1) 121
(2) 144
(3) 169
(4) 225
Q3 - 2024 (27 Jan Shift 1)
If ?? = l i m
?? ? 0
?
v
1 + v 1 + ?? 4
- v 2
?? 4
and ?? = l i m
?? ? 0
?
s i n
2
? ?? v 2 - v 1 + c o s ? ?? , then the value of ?? ?? 3
is :
(1) 36
(2) 32
(3) 25
(4) 30
Q4 - 2024 (27 Jan Shift 2)
If l i m
?? ? 0
?
3 + ?? s i n ? ?? + ?? c o s ? ?? + l o g
?? ? ( 1 - ?? )
3 ta n
2
? ?? =
1
3
, then 2 ?? - ?? is equal to :
(1) 2
Page 2

JEE Mains Previous Year Questions
(2021-2024): Limits and Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
Let { ?? } denote the fractional part of x and ?? ( ?? ) =
c o s
- 1
? ( 1 - { ?? }
2
) s i n
- 1
? ( 1 - { ?? } )
{ ?? } - { ?? }
3
, ?? ? 0. If ?? and R
respectively denotes the left hand limit and the right hand limit of ?? ( ?? ) at ?? = 0, then
32
?? 2
( ?? 2
+ ?? 2
) is equal to
Q2 - 2024 (01 Feb Shift 2)
Let ?? ( ?? ) = {
?? - 1 , ?? is even,
2 ?? , ?? is odd,
?? . If for some ?? ? ?? , ?? ( ?? ( ?? ( ?? ) ) ) = 21, then l i m
?? ? ?? - ? {
| ?? |
3
?? -
[
?? ?? ] }, where [ ?? ] denotes the greatest integer less than or equal to ?? , is equal to :
(1) 121
(2) 144
(3) 169
(4) 225
Q3 - 2024 (27 Jan Shift 1)
If ?? = l i m
?? ? 0
?
v
1 + v 1 + ?? 4
- v 2
?? 4
and ?? = l i m
?? ? 0
?
s i n
2
? ?? v 2 - v 1 + c o s ? ?? , then the value of ?? ?? 3
is :
(1) 36
(2) 32
(3) 25
(4) 30
Q4 - 2024 (27 Jan Shift 2)
If l i m
?? ? 0
?
3 + ?? s i n ? ?? + ?? c o s ? ?? + l o g
?? ? ( 1 - ?? )
3 ta n
2
? ?? =
1
3
, then 2 ?? - ?? is equal to :
(1) 2
(2) 7
(3) 5
(4) 1
Q5 - 2024 (29 Jan Shift 1)
l i m
?? ?
?? 2
? (
1
( ?? -
?? 2
)
2
?
?? 3
(
?? 2
)
3
? c o s ? (
1
?? 3
) ???? ) is equal to
(1)
3 ?? 8

(2)
3 ?? 2
4

(3)
3 ?? 2
8

(4)
3 ?? 4

Q6 - 2024 (29 Jan Shift 2)
Let the slope of the line 45 ?? + 5 ?? + 3 = 0 be 27 ?? 1
+
9 ?? 2
2
for some ?? 1
, ? ?? 2
? ?? . Then
L i m
?? ? 3
? ( ?
3
?? ?
8 ?? 2
3 ?? 2
?? 2
- ?? 2
?? 2
- ?? 1
?? 3
- 3 ?? ???? ) is equal to
Q7 - 2024 (30 Jan Shift 1)
Let f : [ -
?? 2
,
?? 2
] ? R be a differentiable function such that f ( 0 ) =
1
2
, If the l i m
x ? 0
?
x ?
0
x
? f ( t ) dt
e
x
2
- 1
= ?? ,
then 8 ?? 2
is equal to :
(1) 16
(2) 2
(3) 1
(4) 4
Q8 - 2024 (31 Jan Shift 1)
Let ?? be the sum of all coefficients in the expansion of ( 1 - 2 ?? + 2 ?? 2
)
2023
( 3 - 4 ?? 2
+
2 ?? 3
)
2024
and ?? = l i m
?? ? 0
? (
?
0
?? ?
log ? ( 1 + ?? )
?? 2024
+ 1
?? 2
). If the equations cx
2
+ dx + e = 0 and 2 bx
2
+ ax +
4 = 0 have a common root, where ?? , ?? , ?? ? ?? , then ?? : ?? : e equals
Page 3

JEE Mains Previous Year Questions
(2021-2024): Limits and Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
Let { ?? } denote the fractional part of x and ?? ( ?? ) =
c o s
- 1
? ( 1 - { ?? }
2
) s i n
- 1
? ( 1 - { ?? } )
{ ?? } - { ?? }
3
, ?? ? 0. If ?? and R
respectively denotes the left hand limit and the right hand limit of ?? ( ?? ) at ?? = 0, then
32
?? 2
( ?? 2
+ ?? 2
) is equal to
Q2 - 2024 (01 Feb Shift 2)
Let ?? ( ?? ) = {
?? - 1 , ?? is even,
2 ?? , ?? is odd,
?? . If for some ?? ? ?? , ?? ( ?? ( ?? ( ?? ) ) ) = 21, then l i m
?? ? ?? - ? {
| ?? |
3
?? -
[
?? ?? ] }, where [ ?? ] denotes the greatest integer less than or equal to ?? , is equal to :
(1) 121
(2) 144
(3) 169
(4) 225
Q3 - 2024 (27 Jan Shift 1)
If ?? = l i m
?? ? 0
?
v
1 + v 1 + ?? 4
- v 2
?? 4
and ?? = l i m
?? ? 0
?
s i n
2
? ?? v 2 - v 1 + c o s ? ?? , then the value of ?? ?? 3
is :
(1) 36
(2) 32
(3) 25
(4) 30
Q4 - 2024 (27 Jan Shift 2)
If l i m
?? ? 0
?
3 + ?? s i n ? ?? + ?? c o s ? ?? + l o g
?? ? ( 1 - ?? )
3 ta n
2
? ?? =
1
3
, then 2 ?? - ?? is equal to :
(1) 2
(2) 7
(3) 5
(4) 1
Q5 - 2024 (29 Jan Shift 1)
l i m
?? ?
?? 2
? (
1
( ?? -
?? 2
)
2
?
?? 3
(
?? 2
)
3
? c o s ? (
1
?? 3
) ???? ) is equal to
(1)
3 ?? 8

(2)
3 ?? 2
4

(3)
3 ?? 2
8

(4)
3 ?? 4

Q6 - 2024 (29 Jan Shift 2)
Let the slope of the line 45 ?? + 5 ?? + 3 = 0 be 27 ?? 1
+
9 ?? 2
2
for some ?? 1
, ? ?? 2
? ?? . Then
L i m
?? ? 3
? ( ?
3
?? ?
8 ?? 2
3 ?? 2
?? 2
- ?? 2
?? 2
- ?? 1
?? 3
- 3 ?? ???? ) is equal to
Q7 - 2024 (30 Jan Shift 1)
Let f : [ -
?? 2
,
?? 2
] ? R be a differentiable function such that f ( 0 ) =
1
2
, If the l i m
x ? 0
?
x ?
0
x
? f ( t ) dt
e
x
2
- 1
= ?? ,
then 8 ?? 2
is equal to :
(1) 16
(2) 2
(3) 1
(4) 4
Q8 - 2024 (31 Jan Shift 1)
Let ?? be the sum of all coefficients in the expansion of ( 1 - 2 ?? + 2 ?? 2
)
2023
( 3 - 4 ?? 2
+
2 ?? 3
)
2024
and ?? = l i m
?? ? 0
? (
?
0
?? ?
log ? ( 1 + ?? )
?? 2024
+ 1
?? 2
). If the equations cx
2
+ dx + e = 0 and 2 bx
2
+ ax +
4 = 0 have a common root, where ?? , ?? , ?? ? ?? , then ?? : ?? : e equals
(1) 2 : 1 : 4
(2) 4 : 1 : 4
(3) 1 : 2 : 4
(4) 1 : 1 : 4
Q9 - 2024 (31 Jan Shift 1)
l i m
?? ? 0
?
?? 4 s i n ? ?? |
- 2 | s i n ? ?? | - 1
?? 2

(1) is equal to -1
(2) does not exist
(3) is equal to 1
(4) is equal to 2
Q10 - 2024 (31 Jan Shift 2)
Let ? f : ? R ? ( 0 , 8 ) be strictly increasing function such that l i m
?? ? 8
?
?? ( 7 ?? )
?? ( ?? )
= 1. Then, the
value of l i m
?? ? 8
? [
?? ( 5 ?? )
?? ( ?? )
- 1 ] is equal to
(1) 4
(2) 0
(3) 7 / 5
(4) 1
Q11 - 2024 (31 Jan Shift 2)
l i m
?? ? 0
?
?? ?? 2
?? ?? - ?? l o g
?? ? ( 1 + ?? ) + ???? ?? - ?? ?? 2
s i n ? ?? = 1 then 16 ( ?? 2
+ ?? 2
+ ?? 2
) is equal to
Q12 - 2024 (01 Feb Shift 2)
If ?? =
( v ?? + 1 ) ( ?? 2
- v ?? )
?? v ?? + ?? + v ?? +
1
15
( 3 c o s
2
? ?? - 5 ) c o s
3
? ?? , then 96 ?? '
(
?? 6
) is equal to :
Q13 - 2024 (27 Jan Shift 1)
Page 4

JEE Mains Previous Year Questions
(2021-2024): Limits and Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
Let { ?? } denote the fractional part of x and ?? ( ?? ) =
c o s
- 1
? ( 1 - { ?? }
2
) s i n
- 1
? ( 1 - { ?? } )
{ ?? } - { ?? }
3
, ?? ? 0. If ?? and R
respectively denotes the left hand limit and the right hand limit of ?? ( ?? ) at ?? = 0, then
32
?? 2
( ?? 2
+ ?? 2
) is equal to
Q2 - 2024 (01 Feb Shift 2)
Let ?? ( ?? ) = {
?? - 1 , ?? is even,
2 ?? , ?? is odd,
?? . If for some ?? ? ?? , ?? ( ?? ( ?? ( ?? ) ) ) = 21, then l i m
?? ? ?? - ? {
| ?? |
3
?? -
[
?? ?? ] }, where [ ?? ] denotes the greatest integer less than or equal to ?? , is equal to :
(1) 121
(2) 144
(3) 169
(4) 225
Q3 - 2024 (27 Jan Shift 1)
If ?? = l i m
?? ? 0
?
v
1 + v 1 + ?? 4
- v 2
?? 4
and ?? = l i m
?? ? 0
?
s i n
2
? ?? v 2 - v 1 + c o s ? ?? , then the value of ?? ?? 3
is :
(1) 36
(2) 32
(3) 25
(4) 30
Q4 - 2024 (27 Jan Shift 2)
If l i m
?? ? 0
?
3 + ?? s i n ? ?? + ?? c o s ? ?? + l o g
?? ? ( 1 - ?? )
3 ta n
2
? ?? =
1
3
, then 2 ?? - ?? is equal to :
(1) 2
(2) 7
(3) 5
(4) 1
Q5 - 2024 (29 Jan Shift 1)
l i m
?? ?
?? 2
? (
1
( ?? -
?? 2
)
2
?
?? 3
(
?? 2
)
3
? c o s ? (
1
?? 3
) ???? ) is equal to
(1)
3 ?? 8

(2)
3 ?? 2
4

(3)
3 ?? 2
8

(4)
3 ?? 4

Q6 - 2024 (29 Jan Shift 2)
Let the slope of the line 45 ?? + 5 ?? + 3 = 0 be 27 ?? 1
+
9 ?? 2
2
for some ?? 1
, ? ?? 2
? ?? . Then
L i m
?? ? 3
? ( ?
3
?? ?
8 ?? 2
3 ?? 2
?? 2
- ?? 2
?? 2
- ?? 1
?? 3
- 3 ?? ???? ) is equal to
Q7 - 2024 (30 Jan Shift 1)
Let f : [ -
?? 2
,
?? 2
] ? R be a differentiable function such that f ( 0 ) =
1
2
, If the l i m
x ? 0
?
x ?
0
x
? f ( t ) dt
e
x
2
- 1
= ?? ,
then 8 ?? 2
is equal to :
(1) 16
(2) 2
(3) 1
(4) 4
Q8 - 2024 (31 Jan Shift 1)
Let ?? be the sum of all coefficients in the expansion of ( 1 - 2 ?? + 2 ?? 2
)
2023
( 3 - 4 ?? 2
+
2 ?? 3
)
2024
and ?? = l i m
?? ? 0
? (
?
0
?? ?
log ? ( 1 + ?? )
?? 2024
+ 1
?? 2
). If the equations cx
2
+ dx + e = 0 and 2 bx
2
+ ax +
4 = 0 have a common root, where ?? , ?? , ?? ? ?? , then ?? : ?? : e equals
(1) 2 : 1 : 4
(2) 4 : 1 : 4
(3) 1 : 2 : 4
(4) 1 : 1 : 4
Q9 - 2024 (31 Jan Shift 1)
l i m
?? ? 0
?
?? 4 s i n ? ?? |
- 2 | s i n ? ?? | - 1
?? 2

(1) is equal to -1
(2) does not exist
(3) is equal to 1
(4) is equal to 2
Q10 - 2024 (31 Jan Shift 2)
Let ? f : ? R ? ( 0 , 8 ) be strictly increasing function such that l i m
?? ? 8
?
?? ( 7 ?? )
?? ( ?? )
= 1. Then, the
value of l i m
?? ? 8
? [
?? ( 5 ?? )
?? ( ?? )
- 1 ] is equal to
(1) 4
(2) 0
(3) 7 / 5
(4) 1
Q11 - 2024 (31 Jan Shift 2)
l i m
?? ? 0
?
?? ?? 2
?? ?? - ?? l o g
?? ? ( 1 + ?? ) + ???? ?? - ?? ?? 2
s i n ? ?? = 1 then 16 ( ?? 2
+ ?? 2
+ ?? 2
) is equal to
Q12 - 2024 (01 Feb Shift 2)
If ?? =
( v ?? + 1 ) ( ?? 2
- v ?? )
?? v ?? + ?? + v ?? +
1
15
( 3 c o s
2
? ?? - 5 ) c o s
3
? ?? , then 96 ?? '
(
?? 6
) is equal to :
Q13 - 2024 (27 Jan Shift 1)
Let for a differentiable function ?? : ( 0 , 8 ) ? ?? , f ( x ) - f ( y ) = log
e
? (
x
y
) + x - y , ? x , y ? ( 0 , 8 ) .
Then ?
n = 1
20
? f
'
(
1
n
2
) is equal to

Q14 - 2024 (27 Jan Shift 1)
Let ?? ( ?? ) = ?? 3
+ ?? 2
?? '
( 1 ) + ?? ?? ''
( 2 ) + ?? '''
( 3 ) , ?? ? ?? .
Then ?? '
( 10 ) is equal to
Q15 - 2024 (29 Jan Shift 1)
Suppose
?? ( ?? ) =
( 2
?? + 2
- ?? ) t a n ? ?? v t a n
- 1
? ( ?? 2
- ?? + 1 )
( 7 ?? 2
+ 3 ?? + 1 )
3

Then the value of ?? '
( 0 ) is equal to
(1) ??
(2) 0
(3) v ??
(4)
?? 2

Q16 - 2024 (29 Jan Shift 2)
Let ?? = log
?? ? (
1 - ?? 2
1 + ?? 2
) , - 1 < ?? < 1. Then at ?? =
1
2
, the value of 225 ( ?? '
- ?? ''
) is equal to
(1) 732
(2) 746
(3) 742
(4) 736

Q1 (18)
Q2(2)
Page 5

JEE Mains Previous Year Questions
(2021-2024): Limits and Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
Let { ?? } denote the fractional part of x and ?? ( ?? ) =
c o s
- 1
? ( 1 - { ?? }
2
) s i n
- 1
? ( 1 - { ?? } )
{ ?? } - { ?? }
3
, ?? ? 0. If ?? and R
respectively denotes the left hand limit and the right hand limit of ?? ( ?? ) at ?? = 0, then
32
?? 2
( ?? 2
+ ?? 2
) is equal to
Q2 - 2024 (01 Feb Shift 2)
Let ?? ( ?? ) = {
?? - 1 , ?? is even,
2 ?? , ?? is odd,
?? . If for some ?? ? ?? , ?? ( ?? ( ?? ( ?? ) ) ) = 21, then l i m
?? ? ?? - ? {
| ?? |
3
?? -
[
?? ?? ] }, where [ ?? ] denotes the greatest integer less than or equal to ?? , is equal to :
(1) 121
(2) 144
(3) 169
(4) 225
Q3 - 2024 (27 Jan Shift 1)
If ?? = l i m
?? ? 0
?
v
1 + v 1 + ?? 4
- v 2
?? 4
and ?? = l i m
?? ? 0
?
s i n
2
? ?? v 2 - v 1 + c o s ? ?? , then the value of ?? ?? 3
is :
(1) 36
(2) 32
(3) 25
(4) 30
Q4 - 2024 (27 Jan Shift 2)
If l i m
?? ? 0
?
3 + ?? s i n ? ?? + ?? c o s ? ?? + l o g
?? ? ( 1 - ?? )
3 ta n
2
? ?? =
1
3
, then 2 ?? - ?? is equal to :
(1) 2
(2) 7
(3) 5
(4) 1
Q5 - 2024 (29 Jan Shift 1)
l i m
?? ?
?? 2
? (
1
( ?? -
?? 2
)
2
?
?? 3
(
?? 2
)
3
? c o s ? (
1
?? 3
) ???? ) is equal to
(1)
3 ?? 8

(2)
3 ?? 2
4

(3)
3 ?? 2
8

(4)
3 ?? 4

Q6 - 2024 (29 Jan Shift 2)
Let the slope of the line 45 ?? + 5 ?? + 3 = 0 be 27 ?? 1
+
9 ?? 2
2
for some ?? 1
, ? ?? 2
? ?? . Then
L i m
?? ? 3
? ( ?
3
?? ?
8 ?? 2
3 ?? 2
?? 2
- ?? 2
?? 2
- ?? 1
?? 3
- 3 ?? ???? ) is equal to
Q7 - 2024 (30 Jan Shift 1)
Let f : [ -
?? 2
,
?? 2
] ? R be a differentiable function such that f ( 0 ) =
1
2
, If the l i m
x ? 0
?
x ?
0
x
? f ( t ) dt
e
x
2
- 1
= ?? ,
then 8 ?? 2
is equal to :
(1) 16
(2) 2
(3) 1
(4) 4
Q8 - 2024 (31 Jan Shift 1)
Let ?? be the sum of all coefficients in the expansion of ( 1 - 2 ?? + 2 ?? 2
)
2023
( 3 - 4 ?? 2
+
2 ?? 3
)
2024
and ?? = l i m
?? ? 0
? (
?
0
?? ?
log ? ( 1 + ?? )
?? 2024
+ 1
?? 2
). If the equations cx
2
+ dx + e = 0 and 2 bx
2
+ ax +
4 = 0 have a common root, where ?? , ?? , ?? ? ?? , then ?? : ?? : e equals
(1) 2 : 1 : 4
(2) 4 : 1 : 4
(3) 1 : 2 : 4
(4) 1 : 1 : 4
Q9 - 2024 (31 Jan Shift 1)
l i m
?? ? 0
?
?? 4 s i n ? ?? |
- 2 | s i n ? ?? | - 1
?? 2

(1) is equal to -1
(2) does not exist
(3) is equal to 1
(4) is equal to 2
Q10 - 2024 (31 Jan Shift 2)
Let ? f : ? R ? ( 0 , 8 ) be strictly increasing function such that l i m
?? ? 8
?
?? ( 7 ?? )
?? ( ?? )
= 1. Then, the
value of l i m
?? ? 8
? [
?? ( 5 ?? )
?? ( ?? )
- 1 ] is equal to
(1) 4
(2) 0
(3) 7 / 5
(4) 1
Q11 - 2024 (31 Jan Shift 2)
l i m
?? ? 0
?
?? ?? 2
?? ?? - ?? l o g
?? ? ( 1 + ?? ) + ???? ?? - ?? ?? 2
s i n ? ?? = 1 then 16 ( ?? 2
+ ?? 2
+ ?? 2
) is equal to
Q12 - 2024 (01 Feb Shift 2)
If ?? =
( v ?? + 1 ) ( ?? 2
- v ?? )
?? v ?? + ?? + v ?? +
1
15
( 3 c o s
2
? ?? - 5 ) c o s
3
? ?? , then 96 ?? '
(
?? 6
) is equal to :
Q13 - 2024 (27 Jan Shift 1)
Let for a differentiable function ?? : ( 0 , 8 ) ? ?? , f ( x ) - f ( y ) = log
e
? (
x
y
) + x - y , ? x , y ? ( 0 , 8 ) .
Then ?
n = 1
20
? f
'
(
1
n
2
) is equal to

Q14 - 2024 (27 Jan Shift 1)
Let ?? ( ?? ) = ?? 3
+ ?? 2
?? '
( 1 ) + ?? ?? ''
( 2 ) + ?? '''
( 3 ) , ?? ? ?? .
Then ?? '
( 10 ) is equal to
Q15 - 2024 (29 Jan Shift 1)
Suppose
?? ( ?? ) =
( 2
?? + 2
- ?? ) t a n ? ?? v t a n
- 1
? ( ?? 2
- ?? + 1 )
( 7 ?? 2
+ 3 ?? + 1 )
3

Then the value of ?? '
( 0 ) is equal to
(1) ??
(2) 0
(3) v ??
(4)
?? 2

Q16 - 2024 (29 Jan Shift 2)
Let ?? = log
?? ? (
1 - ?? 2
1 + ?? 2
) , - 1 < ?? < 1. Then at ?? =
1
2
, the value of 225 ( ?? '
- ?? ''
) is equal to
(1) 732
(2) 746
(3) 742
(4) 736

Q1 (18)
Q2(2)
Q3 (2)
Q4 (3)
Q5 (3)
Q6 (12)
Q7 (2)
Q8 (4)
Q9 (4)
Q10 (2)
Q11 (81)
Q12 (105) ?
Q13 (2890)
Q14 (202)
Q15 (3)
Q16 (4)
Solutions
Q1
Finding right hand limit
l i m
?? ? 0
+ ? ?? ( ?? ) = l i m
h ? 0
? ?? ( 0 + h )
= l i m
h ? 0
? ?? ( h )
= l i m
h ? 0
?
c o s
- 1
? ( 1 - h
2
) sin
- 1
? ( 1 - h )
h ( 1 - h
2
)

= l i m
h ? 0
?
c o s
- 1
? ( 1 - h
2
)
h
(
sin
- 1
? 1
1
)
Let c o s
- 1
? ( 1 - h
2
) = ?? ? c o s ? ?? = 1 - h
2

=
?? 2
l i m
?? ? 0
?
?? v 1 - c o s ? ??
=
?? 2
l i m
?? ? 0
?
1
v
1 - c o s ? ?? ?? 2

```

## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on Limits and Derivatives: JEE Mains Previous Year Questions (2021-2024) - Mathematics (Maths) for JEE Main & Advanced

 1. What are limits in calculus?
Ans. Limits in calculus refer to the value that a function approaches as the input (or variable) gets arbitrarily close to a particular value. It is used to analyze the behavior of a function near a specific point and determine its continuity, differentiability, and other properties.
 2. How do you find the limit of a function?
Ans. To find the limit of a function, you can evaluate the function at the given value or use algebraic techniques such as factoring, rationalizing, or simplifying. You can also use specific limit laws and theorems to evaluate limits, such as the limit of a sum, difference, product, quotient, or composition of functions.
 3. What is the difference between left-hand limit and right-hand limit?
Ans. The left-hand limit of a function at a specific point is the value that the function approaches as the input approaches that point from the left side. On the other hand, the right-hand limit is the value that the function approaches as the input approaches the point from the right side. If the left-hand limit is equal to the right-hand limit, then the overall limit exists.
 4. What is the derivative of a function?
Ans. The derivative of a function measures the rate at which the function changes with respect to its independent variable. It represents the slope of the tangent line to the graph of the function at a given point. The derivative is calculated using the limit definition of the derivative or through various rules and formulas, such as the power rule, product rule, quotient rule, and chain rule.
 5. How can derivatives be used in real-life applications?
Ans. Derivatives have numerous applications in real-life scenarios. They are used to analyze rates of change, optimize functions, determine the direction of motion, solve optimization problems, predict future behavior based on current trends, approximate values, and solve various physics problems, such as velocity, acceleration, and position. They are also extensively used in economics, engineering, and other fields for modeling and analysis purposes.

## Mathematics (Maths) for JEE Main & Advanced

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