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Limits & Derivatives Class 11 Notes Maths Chapter 12

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 Page 1


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
Page 2


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
Page 3


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
 
?
x
e
x 0
a 1
lim log a
x
?
?
?
?
x 0
log(1 x)
lim 1
x
?
?
?
?
? ?
1 x
x 0
lim 1 x e
?
? ?
? ? s in
cos
d x
x
dx
?
 
? ? cos
s in
d x
x
dx
? ?
 
? ?
2
ta n
se c
d x
x
dx
?
 
? ?
2
cot
cos
d x
ec x
dx
? ?
 
? ? se c
se c ta n
d x
x x
dx
?
? ? cos
cos .cot
d ec x
ec x x
dx
? ?
? ?
1
.
n
n
d x
n x
dx
?
? 
Page 4


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
 
?
x
e
x 0
a 1
lim log a
x
?
?
?
?
x 0
log(1 x)
lim 1
x
?
?
?
?
? ?
1 x
x 0
lim 1 x e
?
? ?
? ? s in
cos
d x
x
dx
?
 
? ? cos
s in
d x
x
dx
? ?
 
? ?
2
ta n
se c
d x
x
dx
?
 
? ?
2
cot
cos
d x
ec x
dx
? ?
 
? ? se c
se c ta n
d x
x x
dx
?
? ? cos
cos .cot
d ec x
ec x x
dx
? ?
? ?
1
.
n
n
d x
n x
dx
?
? 
 
? ?
x
x
d e
e
dx
?
? ?
.log
x
x
d a
a a
dx
?
? ? log 1
e
d x
dx x
?
? ?
0
d constant
dx
?
 
Laws of Logarithm 
? ? ? log log log
e e e
A B AB ? ?
?
log log log
e e e
A
A B
B
? ?
? ?
? ?
? ?
? log log
m
e e
A m A ? 
? log 1 0
a
?
? If log then ? ?
x
B
A x B A
Let y = f(x) be a function defined in some neighbourhood of the 
point ‘a’. Let P(a, f(a)) and Q(a + h, f (a + h)) are two points on 
the graph of f(x) where h is very small and 0 h ? . 
Page 5


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
 
?
x
e
x 0
a 1
lim log a
x
?
?
?
?
x 0
log(1 x)
lim 1
x
?
?
?
?
? ?
1 x
x 0
lim 1 x e
?
? ?
? ? s in
cos
d x
x
dx
?
 
? ? cos
s in
d x
x
dx
? ?
 
? ?
2
ta n
se c
d x
x
dx
?
 
? ?
2
cot
cos
d x
ec x
dx
? ?
 
? ? se c
se c ta n
d x
x x
dx
?
? ? cos
cos .cot
d ec x
ec x x
dx
? ?
? ?
1
.
n
n
d x
n x
dx
?
? 
 
? ?
x
x
d e
e
dx
?
? ?
.log
x
x
d a
a a
dx
?
? ? log 1
e
d x
dx x
?
? ?
0
d constant
dx
?
 
Laws of Logarithm 
? ? ? log log log
e e e
A B AB ? ?
?
log log log
e e e
A
A B
B
? ?
? ?
? ?
? ?
? log log
m
e e
A m A ? 
? log 1 0
a
?
? If log then ? ?
x
B
A x B A
Let y = f(x) be a function defined in some neighbourhood of the 
point ‘a’. Let P(a, f(a)) and Q(a + h, f (a + h)) are two points on 
the graph of f(x) where h is very small and 0 h ? . 
 
Slope of PQ = 
? ? ? ? f a h f a
h
? ?
If 0 h ? , point Q approaches to P and the line PQ becomes a 
tangent to the curve at point P. 
? ? ? ?
0
lim
h
f a h f a
h
?
? ?
 (if exists) is called derivative of f(x) at the 
point ‘a’. It is denoted by f’(a). 
Algebra of derivatives 
?
? ? ? ? ? ? ? ?
.
d d
cf x c f x
dx dx
?
where c is a constant 
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
d d d
f x g x f x g x
dx dx dx
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
. .
d d d
f x g x f x g x g x f x
dx dx dx
? ?
 
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Up next

FAQs on Limits & Derivatives Class 11 Notes Maths Chapter 12

1. What is the concept of limits in calculus?
Ans. In calculus, limits are used to determine the behavior of a function as it approaches a certain value or as the input approaches infinity or negative infinity. It helps us understand how a function behaves near a particular point.
2. How do you find the limit of a function algebraically?
Ans. To find the limit of a function algebraically, you can try to simplify the expression or use algebraic manipulation techniques such as factoring or rationalizing the numerator or denominator. Sometimes, you may need to apply limit laws or use special limit theorems to evaluate the limit.
3. What is the significance of derivatives in calculus?
Ans. Derivatives in calculus represent the rate at which a function is changing at any given point. They provide valuable information about the slope of a function, the direction of its graph, and can be used to find maximum and minimum points. Derivatives are essential in many areas of science, engineering, and economics.
4. How do you find the derivative of a function?
Ans. To find the derivative of a function, you can use differentiation techniques such as the power rule, product rule, quotient rule, or chain rule. These rules allow you to find the rate of change of a function with respect to its independent variable.
5. What is the relationship between limits and derivatives?
Ans. The concept of a derivative is closely related to the concept of a limit. In fact, the derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Limits are used in calculus to study the behavior of functions, while derivatives provide information about the rate of change of a function at a specific point.
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Limits & Derivatives Class 11 Notes Maths Chapter 12

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