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NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

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 Page 1


 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
         
Exercise 2 1 .1 
 
Question 1: 
Evaluate the Given limit: 
x3
lim 3 x
?
? 
Solution 1: 
x3
lim 3 x
?
? = 3 + 3 = 6 
 
 
Question 2: 
Evaluate the Given limit: 
x
22
lim
7
x
? ?
??
?
??
??
 
Solution 2: 
x
22
lim
7
x
? ?
??
?
??
??
= 
22
7
?
??
?
??
??
 
 
 
Question 3: 
Evaluate the Given limit : 
2
x1
lim r ?
?
 
Solution 3: 
2
x1
lim r ?
?
= ? ?
2
1 ?? ? 
 
 
Question 4: 
Evaluate the Given limit: 
x1
43
lim
2
x
x
?
?
?
 
Solution 4: 
x1
43
lim
2
x
x
?
?
?
= 
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
 
 
 
Page 2


 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
         
Exercise 2 1 .1 
 
Question 1: 
Evaluate the Given limit: 
x3
lim 3 x
?
? 
Solution 1: 
x3
lim 3 x
?
? = 3 + 3 = 6 
 
 
Question 2: 
Evaluate the Given limit: 
x
22
lim
7
x
? ?
??
?
??
??
 
Solution 2: 
x
22
lim
7
x
? ?
??
?
??
??
= 
22
7
?
??
?
??
??
 
 
 
Question 3: 
Evaluate the Given limit : 
2
x1
lim r ?
?
 
Solution 3: 
2
x1
lim r ?
?
= ? ?
2
1 ?? ? 
 
 
Question 4: 
Evaluate the Given limit: 
x1
43
lim
2
x
x
?
?
?
 
Solution 4: 
x1
43
lim
2
x
x
?
?
?
= 
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
 
 
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 5: 
Evaluate the Given limit:  
10 5
1
1
lim
1
x
xx
x
??
??
?
 
Solution 5: 
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Question 6: 
Evaluate the Given limit: 
? ?
5
0
11
lim
x
x
x
?
??
 
Solution 6: 
? ?
5
0
11
lim
x
x
x
?
??
 
Put x + 1 = y so that y → 1 as x → 0. 
Accordingly, 
? ?
5
0
11
lim
x
x
x
?
??
 = 
5
1
1
lim
1
x
y
y
?
?
?
 
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
 
 
 
Question 7: 
Evaluate the Given limit: 
2
2
2
3 10
lim
4
x
xx
x
?
??
?
 
Solution 7: 
At x = 2, the value of the given rational function takes the form 
0
0
. 
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
= 
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
 
Page 3


 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
         
Exercise 2 1 .1 
 
Question 1: 
Evaluate the Given limit: 
x3
lim 3 x
?
? 
Solution 1: 
x3
lim 3 x
?
? = 3 + 3 = 6 
 
 
Question 2: 
Evaluate the Given limit: 
x
22
lim
7
x
? ?
??
?
??
??
 
Solution 2: 
x
22
lim
7
x
? ?
??
?
??
??
= 
22
7
?
??
?
??
??
 
 
 
Question 3: 
Evaluate the Given limit : 
2
x1
lim r ?
?
 
Solution 3: 
2
x1
lim r ?
?
= ? ?
2
1 ?? ? 
 
 
Question 4: 
Evaluate the Given limit: 
x1
43
lim
2
x
x
?
?
?
 
Solution 4: 
x1
43
lim
2
x
x
?
?
?
= 
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
 
 
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 5: 
Evaluate the Given limit:  
10 5
1
1
lim
1
x
xx
x
??
??
?
 
Solution 5: 
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Question 6: 
Evaluate the Given limit: 
? ?
5
0
11
lim
x
x
x
?
??
 
Solution 6: 
? ?
5
0
11
lim
x
x
x
?
??
 
Put x + 1 = y so that y → 1 as x → 0. 
Accordingly, 
? ?
5
0
11
lim
x
x
x
?
??
 = 
5
1
1
lim
1
x
y
y
?
?
?
 
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
 
 
 
Question 7: 
Evaluate the Given limit: 
2
2
2
3 10
lim
4
x
xx
x
?
??
?
 
Solution 7: 
At x = 2, the value of the given rational function takes the form 
0
0
. 
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
= 
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
2
35
lim
2
3 2 5
22
11
4
x
x
x
?
?
?
?
?
?
?
?
 
 
 
Question 8: 
Evaluate the Given limit: 
4
2
3
81
lim
2 5 3
x
x
xx
?
?
??
 
Solution 8: 
At x = 2, the value of the given rational function takes the form 
0
0
. 
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
2
4
2
33
2
3
2
3 3 9
81
lim lim
2 5 3 3 2 1
39
lim
21
3 3 3 9
2 3 1
6 18
7
108
7
xx
x
x x x
x
x x x x
xx
x
??
?
? ? ?
?
??
? ? ? ?
??
?
?
? ? ?
?
?
?
?
?
 
 
 
Question 9: 
Evaluate the Given limit: 
0
lim
1
x
ax b
cx
?
?
?
 
Solution 9: 
0
lim
1
x
ax b
cx
?
?
?
= 
? ?
? ?
0
01
ab
b
c
?
?
?
 
 
 
Page 4


 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
         
Exercise 2 1 .1 
 
Question 1: 
Evaluate the Given limit: 
x3
lim 3 x
?
? 
Solution 1: 
x3
lim 3 x
?
? = 3 + 3 = 6 
 
 
Question 2: 
Evaluate the Given limit: 
x
22
lim
7
x
? ?
??
?
??
??
 
Solution 2: 
x
22
lim
7
x
? ?
??
?
??
??
= 
22
7
?
??
?
??
??
 
 
 
Question 3: 
Evaluate the Given limit : 
2
x1
lim r ?
?
 
Solution 3: 
2
x1
lim r ?
?
= ? ?
2
1 ?? ? 
 
 
Question 4: 
Evaluate the Given limit: 
x1
43
lim
2
x
x
?
?
?
 
Solution 4: 
x1
43
lim
2
x
x
?
?
?
= 
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
 
 
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 5: 
Evaluate the Given limit:  
10 5
1
1
lim
1
x
xx
x
??
??
?
 
Solution 5: 
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Question 6: 
Evaluate the Given limit: 
? ?
5
0
11
lim
x
x
x
?
??
 
Solution 6: 
? ?
5
0
11
lim
x
x
x
?
??
 
Put x + 1 = y so that y → 1 as x → 0. 
Accordingly, 
? ?
5
0
11
lim
x
x
x
?
??
 = 
5
1
1
lim
1
x
y
y
?
?
?
 
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
 
 
 
Question 7: 
Evaluate the Given limit: 
2
2
2
3 10
lim
4
x
xx
x
?
??
?
 
Solution 7: 
At x = 2, the value of the given rational function takes the form 
0
0
. 
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
= 
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
2
35
lim
2
3 2 5
22
11
4
x
x
x
?
?
?
?
?
?
?
?
 
 
 
Question 8: 
Evaluate the Given limit: 
4
2
3
81
lim
2 5 3
x
x
xx
?
?
??
 
Solution 8: 
At x = 2, the value of the given rational function takes the form 
0
0
. 
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
2
4
2
33
2
3
2
3 3 9
81
lim lim
2 5 3 3 2 1
39
lim
21
3 3 3 9
2 3 1
6 18
7
108
7
xx
x
x x x
x
x x x x
xx
x
??
?
? ? ?
?
??
? ? ? ?
??
?
?
? ? ?
?
?
?
?
?
 
 
 
Question 9: 
Evaluate the Given limit: 
0
lim
1
x
ax b
cx
?
?
?
 
Solution 9: 
0
lim
1
x
ax b
cx
?
?
?
= 
? ?
? ?
0
01
ab
b
c
?
?
?
 
 
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 10: 
Evaluate the Given limit: 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
 
Solution 10: 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
 
At z = 1, the value of the given function takes the form 
0
0
. 
Put 
1
6
z = x so that z →1 as x → 1. 
Accordingly, 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
 = 
2
1
1
lim
1
x
x
x
?
?
?
 
2
1
2 1 1
1
lim
1
2.1 lim
2
x
nn
n
xa
x
x
xa
na
xa
?
??
?
?
?
?
?? ?
??
??
?
??
?
 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
= 2 
 
 
Question 11: 
Evaluate the Given limit: 
? ?
2
2
1
lim
1
x
ax bx c
cx b a
?
??
??
 , a + b + c ? 0 
Solution 11: 
2
2
1
lim
x
ax bx c
cx bx a
?
??
??
= 
? ? ? ?
? ? ? ?
2
2
11
11
a b c
c b a
??
??
 
=
abc
abc
??
??
 
= 1                          [a + b + c ? 0] 
 
 
Page 5


 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
         
Exercise 2 1 .1 
 
Question 1: 
Evaluate the Given limit: 
x3
lim 3 x
?
? 
Solution 1: 
x3
lim 3 x
?
? = 3 + 3 = 6 
 
 
Question 2: 
Evaluate the Given limit: 
x
22
lim
7
x
? ?
??
?
??
??
 
Solution 2: 
x
22
lim
7
x
? ?
??
?
??
??
= 
22
7
?
??
?
??
??
 
 
 
Question 3: 
Evaluate the Given limit : 
2
x1
lim r ?
?
 
Solution 3: 
2
x1
lim r ?
?
= ? ?
2
1 ?? ? 
 
 
Question 4: 
Evaluate the Given limit: 
x1
43
lim
2
x
x
?
?
?
 
Solution 4: 
x1
43
lim
2
x
x
?
?
?
= 
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
 
 
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 5: 
Evaluate the Given limit:  
10 5
1
1
lim
1
x
xx
x
??
??
?
 
Solution 5: 
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Question 6: 
Evaluate the Given limit: 
? ?
5
0
11
lim
x
x
x
?
??
 
Solution 6: 
? ?
5
0
11
lim
x
x
x
?
??
 
Put x + 1 = y so that y → 1 as x → 0. 
Accordingly, 
? ?
5
0
11
lim
x
x
x
?
??
 = 
5
1
1
lim
1
x
y
y
?
?
?
 
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
 
 
 
Question 7: 
Evaluate the Given limit: 
2
2
2
3 10
lim
4
x
xx
x
?
??
?
 
Solution 7: 
At x = 2, the value of the given rational function takes the form 
0
0
. 
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
= 
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
2
35
lim
2
3 2 5
22
11
4
x
x
x
?
?
?
?
?
?
?
?
 
 
 
Question 8: 
Evaluate the Given limit: 
4
2
3
81
lim
2 5 3
x
x
xx
?
?
??
 
Solution 8: 
At x = 2, the value of the given rational function takes the form 
0
0
. 
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
2
4
2
33
2
3
2
3 3 9
81
lim lim
2 5 3 3 2 1
39
lim
21
3 3 3 9
2 3 1
6 18
7
108
7
xx
x
x x x
x
x x x x
xx
x
??
?
? ? ?
?
??
? ? ? ?
??
?
?
? ? ?
?
?
?
?
?
 
 
 
Question 9: 
Evaluate the Given limit: 
0
lim
1
x
ax b
cx
?
?
?
 
Solution 9: 
0
lim
1
x
ax b
cx
?
?
?
= 
? ?
? ?
0
01
ab
b
c
?
?
?
 
 
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 10: 
Evaluate the Given limit: 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
 
Solution 10: 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
 
At z = 1, the value of the given function takes the form 
0
0
. 
Put 
1
6
z = x so that z →1 as x → 1. 
Accordingly, 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
 = 
2
1
1
lim
1
x
x
x
?
?
?
 
2
1
2 1 1
1
lim
1
2.1 lim
2
x
nn
n
xa
x
x
xa
na
xa
?
??
?
?
?
?
?? ?
??
??
?
??
?
 
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
= 2 
 
 
Question 11: 
Evaluate the Given limit: 
? ?
2
2
1
lim
1
x
ax bx c
cx b a
?
??
??
 , a + b + c ? 0 
Solution 11: 
2
2
1
lim
x
ax bx c
cx bx a
?
??
??
= 
? ? ? ?
? ? ? ?
2
2
11
11
a b c
c b a
??
??
 
=
abc
abc
??
??
 
= 1                          [a + b + c ? 0] 
 
 
 Chapter 2 1 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 12: 
Evaluate the Given limit: 
2
11
2
lim
2
x
x
x
??
?
?
 
Solution 12: 
2
11
2
lim
2
x
x
x
??
?
?
 
At x = –2, the value of the given function takes the form 
0
0
 
Now, 
2
11
2
lim
2
x
x
x
??
?
?
 =  
2
2
2
lim
2
x
x
x
x
??
? ??
??
??
?
 
= 
2
1
lim
2
x
x
??
 
? ?
11
2 2 4
?
??
?
 
 
 
Question 13: 
Evaluate the Given limit: 
0
sin
lim
x
ax
bx
?
 
Solution 13: 
0
sin
lim
x
ax
bx
?
 
At x = 0, the value of the given function takes the form 
0
0
. 
Now, 
0
sin
lim
x
ax
bx
?
 = 
0
sin
lim
x
ax ax
ax bx
?
? 
0
sin
lim
x
ax a
ax b
?
??
??
??
??
 
? ?
0
sin
lim 0 0
ax
a ax
x ax
b ax
?
??
? ? ? ?
??
??
 
0
sin
1 lim 1
y
ay
by
a
b
?
??
? ? ?
??
??
?
 
 
 
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FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

1. What is the concept of limits in calculus?
Ans. In calculus, limits are used to describe the behavior of a function as the input approaches a certain value or as the input goes towards infinity or negative infinity. It helps in determining the value that a function approaches as it gets arbitrarily close to a particular input.
2. How are limits and derivatives related in calculus?
Ans. Limits and derivatives are closely related in calculus. The derivative of a function at a particular point is defined as the limit of the average rate of change of the function as the interval around the point shrinks to zero. In other words, the derivative measures the instantaneous rate of change of a function at a specific point.
3. What is the significance of limits and derivatives in the field of mathematics?
Ans. Limits and derivatives play a crucial role in various branches of mathematics. They are fundamental concepts in calculus and are used to solve problems related to rates of change, optimization, and approximation. Limits and derivatives also find applications in physics, engineering, economics, and other scientific fields.
4. How are limits and derivatives applied in real-life scenarios?
Ans. Limits and derivatives have numerous real-life applications. For example, they are used in physics to calculate velocities and accelerations, in economics to analyze demand and supply functions, in engineering to optimize designs, and in finance to determine rates of return and risk assessments. They help in modeling and solving real-world problems involving change and optimization.
5. What are some common techniques used to evaluate limits and derivatives?
Ans. There are several techniques used to evaluate limits and derivatives, including algebraic manipulation, factoring, rationalizing, L'Hôpital's rule, and Taylor series expansions. These methods help in simplifying expressions or determining the behavior of a function at a particular point or as the input approaches a certain value.
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