Commerce Exam > Commerce Notes > Mathematics (Maths) Class 11 > NCERT Solutions Exercise 12.1: Limits and Derivatives- 2
NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
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Page 1
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
Question 20:
Evaluate the Given limit:
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
a, b, a + b ? 0
Solution 20:
At x = 0, the value of the given function takes the form
0
0
.
Now,
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
=
0
sin
lim
sin
x
ax
ax bx
ax
bx
ax bx
bx
?
??
?
??
??
??
?
??
??
=
? ? ? ?
? ?
0 0 0
0 0 0
sin
lim lim lim
Asx 0 ax 0 and bx 0
sin
lim lim lim
x x x
x x x
ax
ax bx
ax
bx
ax bx
bx
? ? ?
? ? ?
??
??
??
??
? ? ? ?
??
?
??
??
=
? ?
00
00
00
lim lim
sin
lim lim 1
lim lim
xx
xx
xx
ax bx
x
ax bx x
??
??
??
?
??
?
??
?
??
=
? ?
? ?
0
0
lim
lim
x
x
ax bx
ax bx
?
?
?
?
=
0
lim
x ?
(1)
= 1
Question 21:
Evaluate the Given limit: ? ?
0
lim cosec cot
x
xx
?
?
Solution 21:
At x = 0, the value of the given function takes the form ? ? ?
Now,
? ?
0
lim cosec cot
x
xx
?
?
=
0
1 cos
lim
sin sin
x
x
xx
?
??
?
??
??
=
0
1 cos
lim
sin
x
x
x
?
???
??
??
Page 2
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
Question 20:
Evaluate the Given limit:
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
a, b, a + b ? 0
Solution 20:
At x = 0, the value of the given function takes the form
0
0
.
Now,
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
=
0
sin
lim
sin
x
ax
ax bx
ax
bx
ax bx
bx
?
??
?
??
??
??
?
??
??
=
? ? ? ?
? ?
0 0 0
0 0 0
sin
lim lim lim
Asx 0 ax 0 and bx 0
sin
lim lim lim
x x x
x x x
ax
ax bx
ax
bx
ax bx
bx
? ? ?
? ? ?
??
??
??
??
? ? ? ?
??
?
??
??
=
? ?
00
00
00
lim lim
sin
lim lim 1
lim lim
xx
xx
xx
ax bx
x
ax bx x
??
??
??
?
??
?
??
?
??
=
? ?
? ?
0
0
lim
lim
x
x
ax bx
ax bx
?
?
?
?
=
0
lim
x ?
(1)
= 1
Question 21:
Evaluate the Given limit: ? ?
0
lim cosec cot
x
xx
?
?
Solution 21:
At x = 0, the value of the given function takes the form ? ? ?
Now,
? ?
0
lim cosec cot
x
xx
?
?
=
0
1 cos
lim
sin sin
x
x
xx
?
??
?
??
??
=
0
1 cos
lim
sin
x
x
x
?
???
??
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
=
0
1 cos
lim
sin
x
x
x
x
x
?
???
??
??
??
??
??
=
0
0
1 cos
lim
sin
lim
x
x
x
x
x
x
?
?
?
=
0
1
00
1 cos sin
lim 0and lim 1
xx
xx
xx
??
???
??
??
??
= 0
Question 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
Solution 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
At x =
2
?
, the value of the given function takes the form
0
0
Now, put So that x , 0
22
x y y
??
? ? ? ?
0
2
tan2
tan2 2
lim lim
2
y
x
y
x
y
x
?
?
? ?
?
??
?
??
??
??
?
=
? ?
0
tan 2
lim
y
y
y
?
?
?
=
? ?
0
tan2
lim tan 2 tan2
y
y
yy
y
?
?
?? ??
??
=
0
sin2
lim
cos2
y
y
yy
?
=
0
sin2 2
lim
2 cos2
y
y
yy
?
??
?
??
??
Page 3
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
Question 20:
Evaluate the Given limit:
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
a, b, a + b ? 0
Solution 20:
At x = 0, the value of the given function takes the form
0
0
.
Now,
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
=
0
sin
lim
sin
x
ax
ax bx
ax
bx
ax bx
bx
?
??
?
??
??
??
?
??
??
=
? ? ? ?
? ?
0 0 0
0 0 0
sin
lim lim lim
Asx 0 ax 0 and bx 0
sin
lim lim lim
x x x
x x x
ax
ax bx
ax
bx
ax bx
bx
? ? ?
? ? ?
??
??
??
??
? ? ? ?
??
?
??
??
=
? ?
00
00
00
lim lim
sin
lim lim 1
lim lim
xx
xx
xx
ax bx
x
ax bx x
??
??
??
?
??
?
??
?
??
=
? ?
? ?
0
0
lim
lim
x
x
ax bx
ax bx
?
?
?
?
=
0
lim
x ?
(1)
= 1
Question 21:
Evaluate the Given limit: ? ?
0
lim cosec cot
x
xx
?
?
Solution 21:
At x = 0, the value of the given function takes the form ? ? ?
Now,
? ?
0
lim cosec cot
x
xx
?
?
=
0
1 cos
lim
sin sin
x
x
xx
?
??
?
??
??
=
0
1 cos
lim
sin
x
x
x
?
???
??
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
=
0
1 cos
lim
sin
x
x
x
x
x
?
???
??
??
??
??
??
=
0
0
1 cos
lim
sin
lim
x
x
x
x
x
x
?
?
?
=
0
1
00
1 cos sin
lim 0and lim 1
xx
xx
xx
??
???
??
??
??
= 0
Question 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
Solution 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
At x =
2
?
, the value of the given function takes the form
0
0
Now, put So that x , 0
22
x y y
??
? ? ? ?
0
2
tan2
tan2 2
lim lim
2
y
x
y
x
y
x
?
?
? ?
?
??
?
??
??
??
?
=
? ?
0
tan 2
lim
y
y
y
?
?
?
=
? ?
0
tan2
lim tan 2 tan2
y
y
yy
y
?
?
?? ??
??
=
0
sin2
lim
cos2
y
y
yy
?
=
0
sin2 2
lim
2 cos2
y
y
yy
?
??
?
??
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
=
2 0 0
sin2 2
lim lim
2 cos2
yy
y
yy
??
? ? ? ?
?
? ? ? ?
? ? ? ?
? ?
0 2 0 yy ? ? ?
=
2
1
cos0
?
0
sin
lim 1
x
x
x
?
??
?
??
??
=
2
1
1
?
= 2
Question 23:
Find
0
lim
x ?
f(x) and
1
lim
x ?
f(x), where f(x) =
? ?
2 3, 0
3 1, 0
xx
xx
?? ?
?
?
??
?
?
Solution 23:
The given function is
f(x) =
? ?
2 3, 0
3 1, 0
xx
xx
?? ?
?
?
??
?
?
? ? ? ? ? ?
0 0
lim lim 2 3 2 0 3 3
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
0 0
lim lim3 1 3 0 1 3
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
0 00
lim lim lim 3
x xx
f x f x f x
??
? ??
? ? ? ?
? ? ? ? ? ?
1 1
lim lim3 1 3 1 1 6
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
1 11
lim lim lim 6
x xx
f x f x f x
??
? ??
? ? ? ?
Question 24:
Find ? ?
1
lim
x
fx
?
, when f(x) =
2
2
1 , 1
1 , 1
xx
xx
???
?
?
? ? ?
?
?
Solution 24:
The given function is
f(x) =
2
2
1, 1
1, 1
xx
xx
???
?
?
? ? ?
?
?
? ?
22
1 1
lim lim 1 1 1 1 1 0
x x
f x x
?
? ?
?? ? ? ? ? ? ? ?
??
Page 4
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
Question 20:
Evaluate the Given limit:
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
a, b, a + b ? 0
Solution 20:
At x = 0, the value of the given function takes the form
0
0
.
Now,
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
=
0
sin
lim
sin
x
ax
ax bx
ax
bx
ax bx
bx
?
??
?
??
??
??
?
??
??
=
? ? ? ?
? ?
0 0 0
0 0 0
sin
lim lim lim
Asx 0 ax 0 and bx 0
sin
lim lim lim
x x x
x x x
ax
ax bx
ax
bx
ax bx
bx
? ? ?
? ? ?
??
??
??
??
? ? ? ?
??
?
??
??
=
? ?
00
00
00
lim lim
sin
lim lim 1
lim lim
xx
xx
xx
ax bx
x
ax bx x
??
??
??
?
??
?
??
?
??
=
? ?
? ?
0
0
lim
lim
x
x
ax bx
ax bx
?
?
?
?
=
0
lim
x ?
(1)
= 1
Question 21:
Evaluate the Given limit: ? ?
0
lim cosec cot
x
xx
?
?
Solution 21:
At x = 0, the value of the given function takes the form ? ? ?
Now,
? ?
0
lim cosec cot
x
xx
?
?
=
0
1 cos
lim
sin sin
x
x
xx
?
??
?
??
??
=
0
1 cos
lim
sin
x
x
x
?
???
??
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
=
0
1 cos
lim
sin
x
x
x
x
x
?
???
??
??
??
??
??
=
0
0
1 cos
lim
sin
lim
x
x
x
x
x
x
?
?
?
=
0
1
00
1 cos sin
lim 0and lim 1
xx
xx
xx
??
???
??
??
??
= 0
Question 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
Solution 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
At x =
2
?
, the value of the given function takes the form
0
0
Now, put So that x , 0
22
x y y
??
? ? ? ?
0
2
tan2
tan2 2
lim lim
2
y
x
y
x
y
x
?
?
? ?
?
??
?
??
??
??
?
=
? ?
0
tan 2
lim
y
y
y
?
?
?
=
? ?
0
tan2
lim tan 2 tan2
y
y
yy
y
?
?
?? ??
??
=
0
sin2
lim
cos2
y
y
yy
?
=
0
sin2 2
lim
2 cos2
y
y
yy
?
??
?
??
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
=
2 0 0
sin2 2
lim lim
2 cos2
yy
y
yy
??
? ? ? ?
?
? ? ? ?
? ? ? ?
? ?
0 2 0 yy ? ? ?
=
2
1
cos0
?
0
sin
lim 1
x
x
x
?
??
?
??
??
=
2
1
1
?
= 2
Question 23:
Find
0
lim
x ?
f(x) and
1
lim
x ?
f(x), where f(x) =
? ?
2 3, 0
3 1, 0
xx
xx
?? ?
?
?
??
?
?
Solution 23:
The given function is
f(x) =
? ?
2 3, 0
3 1, 0
xx
xx
?? ?
?
?
??
?
?
? ? ? ? ? ?
0 0
lim lim 2 3 2 0 3 3
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
0 0
lim lim3 1 3 0 1 3
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
0 00
lim lim lim 3
x xx
f x f x f x
??
? ??
? ? ? ?
? ? ? ? ? ?
1 1
lim lim3 1 3 1 1 6
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
1 11
lim lim lim 6
x xx
f x f x f x
??
? ??
? ? ? ?
Question 24:
Find ? ?
1
lim
x
fx
?
, when f(x) =
2
2
1 , 1
1 , 1
xx
xx
???
?
?
? ? ?
?
?
Solution 24:
The given function is
f(x) =
2
2
1, 1
1, 1
xx
xx
???
?
?
? ? ?
?
?
? ?
22
1 1
lim lim 1 1 1 1 1 0
x x
f x x
?
? ?
?? ? ? ? ? ? ? ?
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
22
1 1
lim lim 1 1 1 1 1 2
x x
f x x
?
? ?
?? ? ? ? ? ? ? ? ? ? ? ?
??
It is observed that ? ?
1
lim
x
fx
?
?
? ? ?
1
lim
x
fx
?
?
.
Hence, ? ?
1
lim
x
fx
?
does not exist.
Question 25:
Evaluate ? ?
0
lim
x
fx
?
, where f(x) =
,0
0, 0
x
x
x
x
?
? ?
?
?
?
?
Solution 25:
The given function is
f(x) =
,0
0, 0
x
x
x
x
?
? ?
?
?
?
?
? ?
00
lim lim
xx
x
fx
x
??
??
??
?
??
??
=
0
lim
x
x
x
?
? ??
??
??
[When x is negative, x = -x]
= ? ?
0
lim 1
x ?
?
= -1
? ?
00
lim lim
xx
x
fx
x
??
??
??
?
??
??
=
0
lim
x
x
x
?
??
??
??
[When x is positive, x = x]
= ? ?
0
lim 1
x ?
= 1
It is observed that ? ?
0
lim
x
fx
?
?
? ? ?
0
lim
x
fx
?
?
.
Hence, ? ?
0
lim
x
fx
?
does not exist.
Page 5
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
Question 20:
Evaluate the Given limit:
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
a, b, a + b ? 0
Solution 20:
At x = 0, the value of the given function takes the form
0
0
.
Now,
0
sin
lim
sin
x
ax bx
ax bx
?
?
?
=
0
sin
lim
sin
x
ax
ax bx
ax
bx
ax bx
bx
?
??
?
??
??
??
?
??
??
=
? ? ? ?
? ?
0 0 0
0 0 0
sin
lim lim lim
Asx 0 ax 0 and bx 0
sin
lim lim lim
x x x
x x x
ax
ax bx
ax
bx
ax bx
bx
? ? ?
? ? ?
??
??
??
??
? ? ? ?
??
?
??
??
=
? ?
00
00
00
lim lim
sin
lim lim 1
lim lim
xx
xx
xx
ax bx
x
ax bx x
??
??
??
?
??
?
??
?
??
=
? ?
? ?
0
0
lim
lim
x
x
ax bx
ax bx
?
?
?
?
=
0
lim
x ?
(1)
= 1
Question 21:
Evaluate the Given limit: ? ?
0
lim cosec cot
x
xx
?
?
Solution 21:
At x = 0, the value of the given function takes the form ? ? ?
Now,
? ?
0
lim cosec cot
x
xx
?
?
=
0
1 cos
lim
sin sin
x
x
xx
?
??
?
??
??
=
0
1 cos
lim
sin
x
x
x
?
???
??
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
=
0
1 cos
lim
sin
x
x
x
x
x
?
???
??
??
??
??
??
=
0
0
1 cos
lim
sin
lim
x
x
x
x
x
x
?
?
?
=
0
1
00
1 cos sin
lim 0and lim 1
xx
xx
xx
??
???
??
??
??
= 0
Question 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
Solution 22:
2
tan 2
lim
2
x
x
x
?
?
?
?
At x =
2
?
, the value of the given function takes the form
0
0
Now, put So that x , 0
22
x y y
??
? ? ? ?
0
2
tan2
tan2 2
lim lim
2
y
x
y
x
y
x
?
?
? ?
?
??
?
??
??
??
?
=
? ?
0
tan 2
lim
y
y
y
?
?
?
=
? ?
0
tan2
lim tan 2 tan2
y
y
yy
y
?
?
?? ??
??
=
0
sin2
lim
cos2
y
y
yy
?
=
0
sin2 2
lim
2 cos2
y
y
yy
?
??
?
??
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
=
2 0 0
sin2 2
lim lim
2 cos2
yy
y
yy
??
? ? ? ?
?
? ? ? ?
? ? ? ?
? ?
0 2 0 yy ? ? ?
=
2
1
cos0
?
0
sin
lim 1
x
x
x
?
??
?
??
??
=
2
1
1
?
= 2
Question 23:
Find
0
lim
x ?
f(x) and
1
lim
x ?
f(x), where f(x) =
? ?
2 3, 0
3 1, 0
xx
xx
?? ?
?
?
??
?
?
Solution 23:
The given function is
f(x) =
? ?
2 3, 0
3 1, 0
xx
xx
?? ?
?
?
??
?
?
? ? ? ? ? ?
0 0
lim lim 2 3 2 0 3 3
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
0 0
lim lim3 1 3 0 1 3
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
0 00
lim lim lim 3
x xx
f x f x f x
??
? ??
? ? ? ?
? ? ? ? ? ?
1 1
lim lim3 1 3 1 1 6
x x
f x x
?
? ?
? ? ? ? ?
? ? ? ? ? ?
1 11
lim lim lim 6
x xx
f x f x f x
??
? ??
? ? ? ?
Question 24:
Find ? ?
1
lim
x
fx
?
, when f(x) =
2
2
1 , 1
1 , 1
xx
xx
???
?
?
? ? ?
?
?
Solution 24:
The given function is
f(x) =
2
2
1, 1
1, 1
xx
xx
???
?
?
? ? ?
?
?
? ?
22
1 1
lim lim 1 1 1 1 1 0
x x
f x x
?
? ?
?? ? ? ? ? ? ? ?
??
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
22
1 1
lim lim 1 1 1 1 1 2
x x
f x x
?
? ?
?? ? ? ? ? ? ? ? ? ? ? ?
??
It is observed that ? ?
1
lim
x
fx
?
?
? ? ?
1
lim
x
fx
?
?
.
Hence, ? ?
1
lim
x
fx
?
does not exist.
Question 25:
Evaluate ? ?
0
lim
x
fx
?
, where f(x) =
,0
0, 0
x
x
x
x
?
? ?
?
?
?
?
Solution 25:
The given function is
f(x) =
,0
0, 0
x
x
x
x
?
? ?
?
?
?
?
? ?
00
lim lim
xx
x
fx
x
??
??
??
?
??
??
=
0
lim
x
x
x
?
? ??
??
??
[When x is negative, x = -x]
= ? ?
0
lim 1
x ?
?
= -1
? ?
00
lim lim
xx
x
fx
x
??
??
??
?
??
??
=
0
lim
x
x
x
?
??
??
??
[When x is positive, x = x]
= ? ?
0
lim 1
x ?
= 1
It is observed that ? ?
0
lim
x
fx
?
?
? ? ?
0
lim
x
fx
?
?
.
Hence, ? ?
0
lim
x
fx
?
does not exist.
Class XI Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
Question 26:
Find ? ?
0
lim
x
fx
?
, when f(x) =
,0
0, 0
x
x
x
x
?
?
?
?
?
?
?
Solution 26:
The given function is
f(x) =
,0
0, 0
x
x
x
x
?
?
?
?
?
?
?
? ?
00
lim lim
xx
x
fx
x
??
??
??
?
??
??
=
0
lim
x
x
x
?
??
??
?
??
[When x < 0, x = -x]
= ? ?
0
lim 1
x ?
?
= -1
? ?
00
lim lim
xx
x
fx
x
??
??
??
?
??
??
=
0
lim
x
x
x
?
??
??
??
[When x > 0, x = x]
= ? ?
0
lim 1
x ?
= 1
It is observed that ? ? ? ?
00
lim lim
xx
f x f x
??
??
? .
Hence, ? ?
0
lim
x
fx
?
does not exist.
Question 27:
Find ? ?
5
lim
x
fx
?
, where f(x) = x - 5
Solution 27:
The given function is f(x) = x - 5
? ? ? ?
55
lim lim 5
xx
f x x
??
??
??
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FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| 1. What are the basic concepts of limits and derivatives in calculus? |  |
Ans. Limits and derivatives are fundamental concepts in calculus. A limit is the value that a function approaches as the input approaches a certain value. Derivatives, on the other hand, represent the rate of change of a function at a specific point.
| 2. How are limits and derivatives used in real-world applications? | |
Ans. Limits and derivatives are used in various real-world applications, such as calculating velocities, accelerations, growth rates, and optimization problems in fields like physics, engineering, economics, and biology.
| 3. What is the difference between a one-sided limit and a two-sided limit? | |
Ans. A one-sided limit approaches a specific value from only one side of the function, either from the left or the right. A two-sided limit, on the other hand, approaches the same value from both sides of the function.
| 4. How do you find the derivative of a function using the first principles method? | |
Ans. To find the derivative of a function using the first principles method, you need to calculate the limit of the difference quotient as the change in the input approaches zero. This process involves finding the slope of the tangent line to the function at a specific point.
| 5. What are some common rules and formulas for finding derivatives of functions? | |
Ans. Some common rules and formulas for finding derivatives include the power rule, product rule, quotient rule, chain rule, and rules for trigonometric, exponential, and logarithmic functions. These rules help simplify the process of finding derivatives for various types of functions.
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