Important Formulas: Limits & Derivatives

# Important Limits & Derivatives Formulas for JEE and NEET

``` Page 1

Page # 8
LIMIT OF FUNCTION
1. Limit of a function f(x) is said to exist as  x ? a when,
?
?
0 h
Limit f (a ? h)      =
?
?
0 h
Limit  f (a + h) =  some finite value M.
(Left hand limit) (Right hand limit)
2. Indeterminant Forms:

0
0
,
?
?
, 0 ? ? ?, ??? ?????º, 0º, ?and 1
?
.
3. Standard Limits:
0 x
Limit
?
x
x sin
= 0 x
Limit
?
x
x tan
= 0 x
Limit
?
x
x tan
1 ?
=
0 x
Limit
?
x
x sin
1 ?
=
0 x
Limit
?

x
1 e
x
?
=
0 x
Limit
?

x
) x 1 ( n ? ?
= 1
0 x
Limit
?
(1 + x)
1/x
=
? ? x
Limit

x
x
1
1 ?
?
?
?
?
?
?
= e,
0 x
Limit
?
x
1 a
x
?
= log
e
a, a > 0,
a x
Limit
?

a x
a x
n n
?
?
= na
n – 1
.
4. Limits Using Expansion
(i) 0 a .........
! 3
a ln x
! 2
a ln x
! 1
a ln x
1 a
33 22
x
? ??? ??
(ii) ......
! 3
x
! 2
x
! 1
x
1 e
3 2
x
? ? ? ? ?
(iii) ln (1+x) = 1 x 1 for .........
4
x
3
x
2
x
x
432
? ? ? ??? ?
(iv) .....
! 7
x
! 5
x
! 3
x
x x sin
7 5 3
?? ?? ?
Page 2

Page # 8
LIMIT OF FUNCTION
1. Limit of a function f(x) is said to exist as  x ? a when,
?
?
0 h
Limit f (a ? h)      =
?
?
0 h
Limit  f (a + h) =  some finite value M.
(Left hand limit) (Right hand limit)
2. Indeterminant Forms:

0
0
,
?
?
, 0 ? ? ?, ??? ?????º, 0º, ?and 1
?
.
3. Standard Limits:
0 x
Limit
?
x
x sin
= 0 x
Limit
?
x
x tan
= 0 x
Limit
?
x
x tan
1 ?
=
0 x
Limit
?
x
x sin
1 ?
=
0 x
Limit
?

x
1 e
x
?
=
0 x
Limit
?

x
) x 1 ( n ? ?
= 1
0 x
Limit
?
(1 + x)
1/x
=
? ? x
Limit

x
x
1
1 ?
?
?
?
?
?
?
= e,
0 x
Limit
?
x
1 a
x
?
= log
e
a, a > 0,
a x
Limit
?

a x
a x
n n
?
?
= na
n – 1
.
4. Limits Using Expansion
(i) 0 a .........
! 3
a ln x
! 2
a ln x
! 1
a ln x
1 a
33 22
x
? ??? ??
(ii) ......
! 3
x
! 2
x
! 1
x
1 e
3 2
x
? ? ? ? ?
(iii) ln (1+x) = 1 x 1 for .........
4
x
3
x
2
x
x
432
? ? ? ??? ?
(iv) .....
! 7
x
! 5
x
! 3
x
x x sin
7 5 3
?? ?? ?
Page # 9
(v) .....
! 6
x
! 4
x
! 2
x
1 x cos
642
??? ? ?
(vi) tan x = ......
15
x 2
3
x
x
5 3
? ? ?
(vii) for |x| < 1, n ? R (1 + x)
n
= 1 + nx +
2 . 1
) 1 n ( n ?
x
2
+
3. 2. 1
) 2 n )( 1 n ( n ? ?
x
3
+ ............ ?
5. Limits of form 1
?
, 0
0
, ?
0
Also for (1)
?
type of  problems we can use following rules.
0 x
lim
?
(1 + x)
1/x
= e,
a x
lim
?
[f(x)]
g(x)
,
where f(x) ? 1   ;    g(x) ?? ??  as x ? a  =
)x ( g ] 1 )x ( f [ lim
a x
e
?
?
6. Sandwich Theorem or Squeeze Play Theorem:
If  f(x) ? g(x) ? h(x) ? x   &
a x
Limit
?
f(x) =
?
=
a x
Limit
?
h(x) then
a x
Limit
?
g(x) =
?
.
Page 3

Page # 8
LIMIT OF FUNCTION
1. Limit of a function f(x) is said to exist as  x ? a when,
?
?
0 h
Limit f (a ? h)      =
?
?
0 h
Limit  f (a + h) =  some finite value M.
(Left hand limit) (Right hand limit)
2. Indeterminant Forms:

0
0
,
?
?
, 0 ? ? ?, ??? ?????º, 0º, ?and 1
?
.
3. Standard Limits:
0 x
Limit
?
x
x sin
= 0 x
Limit
?
x
x tan
= 0 x
Limit
?
x
x tan
1 ?
=
0 x
Limit
?
x
x sin
1 ?
=
0 x
Limit
?

x
1 e
x
?
=
0 x
Limit
?

x
) x 1 ( n ? ?
= 1
0 x
Limit
?
(1 + x)
1/x
=
? ? x
Limit

x
x
1
1 ?
?
?
?
?
?
?
= e,
0 x
Limit
?
x
1 a
x
?
= log
e
a, a > 0,
a x
Limit
?

a x
a x
n n
?
?
= na
n – 1
.
4. Limits Using Expansion
(i) 0 a .........
! 3
a ln x
! 2
a ln x
! 1
a ln x
1 a
33 22
x
? ??? ??
(ii) ......
! 3
x
! 2
x
! 1
x
1 e
3 2
x
? ? ? ? ?
(iii) ln (1+x) = 1 x 1 for .........
4
x
3
x
2
x
x
432
? ? ? ??? ?
(iv) .....
! 7
x
! 5
x
! 3
x
x x sin
7 5 3
?? ?? ?
Page # 9
(v) .....
! 6
x
! 4
x
! 2
x
1 x cos
642
??? ? ?
(vi) tan x = ......
15
x 2
3
x
x
5 3
? ? ?
(vii) for |x| < 1, n ? R (1 + x)
n
= 1 + nx +
2 . 1
) 1 n ( n ?
x
2
+
3. 2. 1
) 2 n )( 1 n ( n ? ?
x
3
+ ............ ?
5. Limits of form 1
?
, 0
0
, ?
0
Also for (1)
?
type of  problems we can use following rules.
0 x
lim
?
(1 + x)
1/x
= e,
a x
lim
?
[f(x)]
g(x)
,
where f(x) ? 1   ;    g(x) ?? ??  as x ? a  =
)x ( g ] 1 )x ( f [ lim
a x
e
?
?
6. Sandwich Theorem or Squeeze Play Theorem:
If  f(x) ? g(x) ? h(x) ? x   &
a x
Limit
?
f(x) =
?
=
a x
Limit
?
h(x) then
a x
Limit
?
g(x) =
?
.
Page # 9
METHOD OF DIFFERENTIATION
1. Differentiation of some elementary functions
1.
dx
d
(x
n
) = nx
n – 1
2.
dx
d
(a
x
) = a
x
?n a
3.
dx
d
( ?n |x|) =
x
1
4. ?
dx
d
(log
a
x) =
a n x
1
?
5.
dx
d
(sin x) = cos x 6.
dx
d
(cos x) = – sin x
7.
dx
d
(sec x) = sec x  tan x 8.
dx
d
(cosec x) = – cosec x cot x
9.
dx
d
(tan x) = sec
2
x 10.
dx
d
(cot x) = – cosec
2
x
Page 4

Page # 8
LIMIT OF FUNCTION
1. Limit of a function f(x) is said to exist as  x ? a when,
?
?
0 h
Limit f (a ? h)      =
?
?
0 h
Limit  f (a + h) =  some finite value M.
(Left hand limit) (Right hand limit)
2. Indeterminant Forms:

0
0
,
?
?
, 0 ? ? ?, ??? ?????º, 0º, ?and 1
?
.
3. Standard Limits:
0 x
Limit
?
x
x sin
= 0 x
Limit
?
x
x tan
= 0 x
Limit
?
x
x tan
1 ?
=
0 x
Limit
?
x
x sin
1 ?
=
0 x
Limit
?

x
1 e
x
?
=
0 x
Limit
?

x
) x 1 ( n ? ?
= 1
0 x
Limit
?
(1 + x)
1/x
=
? ? x
Limit

x
x
1
1 ?
?
?
?
?
?
?
= e,
0 x
Limit
?
x
1 a
x
?
= log
e
a, a > 0,
a x
Limit
?

a x
a x
n n
?
?
= na
n – 1
.
4. Limits Using Expansion
(i) 0 a .........
! 3
a ln x
! 2
a ln x
! 1
a ln x
1 a
33 22
x
? ??? ??
(ii) ......
! 3
x
! 2
x
! 1
x
1 e
3 2
x
? ? ? ? ?
(iii) ln (1+x) = 1 x 1 for .........
4
x
3
x
2
x
x
432
? ? ? ??? ?
(iv) .....
! 7
x
! 5
x
! 3
x
x x sin
7 5 3
?? ?? ?
Page # 9
(v) .....
! 6
x
! 4
x
! 2
x
1 x cos
642
??? ? ?
(vi) tan x = ......
15
x 2
3
x
x
5 3
? ? ?
(vii) for |x| < 1, n ? R (1 + x)
n
= 1 + nx +
2 . 1
) 1 n ( n ?
x
2
+
3. 2. 1
) 2 n )( 1 n ( n ? ?
x
3
+ ............ ?
5. Limits of form 1
?
, 0
0
, ?
0
Also for (1)
?
type of  problems we can use following rules.
0 x
lim
?
(1 + x)
1/x
= e,
a x
lim
?
[f(x)]
g(x)
,
where f(x) ? 1   ;    g(x) ?? ??  as x ? a  =
)x ( g ] 1 )x ( f [ lim
a x
e
?
?
6. Sandwich Theorem or Squeeze Play Theorem:
If  f(x) ? g(x) ? h(x) ? x   &
a x
Limit
?
f(x) =
?
=
a x
Limit
?
h(x) then
a x
Limit
?
g(x) =
?
.
Page # 9
METHOD OF DIFFERENTIATION
1. Differentiation of some elementary functions
1.
dx
d
(x
n
) = nx
n – 1
2.
dx
d
(a
x
) = a
x
?n a
3.
dx
d
( ?n |x|) =
x
1
4. ?
dx
d
(log
a
x) =
a n x
1
?
5.
dx
d
(sin x) = cos x 6.
dx
d
(cos x) = – sin x
7.
dx
d
(sec x) = sec x  tan x 8.
dx
d
(cosec x) = – cosec x cot x
9.
dx
d
(tan x) = sec
2
x 10.
dx
d
(cot x) = – cosec
2
x
Page # 10
2.
1.
dx
d
(f ± g) = f ?(x) ± g ?(x) 2.
dx
d
(k f(x)) = k
dx
d
f(x)
3.
dx
d
(f(x) . g(x)) = f(x) g ?(x) + g(x) f ?(x)
4.
dx
d

?
?
?
?
?
?
?
?
) x ( g
) x ( f
=
) x ( g
)x ( g )x ( f ) x ( f )x ( g
2
? ? ?
5.
dx
d
(f(g(x))) =  f ?(g(x)) g ?(x)
Derivative Of Inverse Trigonometric Functions.
dx
x sin d
1 –
=
2
x 1
1
?
,
dx
x cos d
1 –
= –
2
x 1
1
?
, for – 1 < x < 1.
dx
x tan d
1 –
=
2
x 1
1
?
,
dx
x cot d
1 –
= –
2
x 1
1
?
(x ? R)
dx
x sec d
1 –
=
1 x | x |
1
2
?
,
dx
x ec cos d
1 –
= –
1 x | x |
1
2
?
, for x ? (– ?, – 1) ? (1, ?)
3. Differentiation using substitution
Following substitutions are normally used to simplify these expression.
(i)
2 2
a x ?
by substituting    x = a tan ?, where   –
2
?
< ? ? ? ?
2
?
(ii)
2 2
x a ?
by substituting    x = a sin ?, where   –
2
?
? ? ? ? ?
2
?
(iii)
2 2
a x ?
by substituting    x = a sec ?, where ? ? ? ?[0, ?], ? ? ?
2
?
(iv)
x a
a x
?
?
by substituting      x = a cos ?, where ? ? ? ?(0, ?].
Page 5

Page # 8
LIMIT OF FUNCTION
1. Limit of a function f(x) is said to exist as  x ? a when,
?
?
0 h
Limit f (a ? h)      =
?
?
0 h
Limit  f (a + h) =  some finite value M.
(Left hand limit) (Right hand limit)
2. Indeterminant Forms:

0
0
,
?
?
, 0 ? ? ?, ??? ?????º, 0º, ?and 1
?
.
3. Standard Limits:
0 x
Limit
?
x
x sin
= 0 x
Limit
?
x
x tan
= 0 x
Limit
?
x
x tan
1 ?
=
0 x
Limit
?
x
x sin
1 ?
=
0 x
Limit
?

x
1 e
x
?
=
0 x
Limit
?

x
) x 1 ( n ? ?
= 1
0 x
Limit
?
(1 + x)
1/x
=
? ? x
Limit

x
x
1
1 ?
?
?
?
?
?
?
= e,
0 x
Limit
?
x
1 a
x
?
= log
e
a, a > 0,
a x
Limit
?

a x
a x
n n
?
?
= na
n – 1
.
4. Limits Using Expansion
(i) 0 a .........
! 3
a ln x
! 2
a ln x
! 1
a ln x
1 a
33 22
x
? ??? ??
(ii) ......
! 3
x
! 2
x
! 1
x
1 e
3 2
x
? ? ? ? ?
(iii) ln (1+x) = 1 x 1 for .........
4
x
3
x
2
x
x
432
? ? ? ??? ?
(iv) .....
! 7
x
! 5
x
! 3
x
x x sin
7 5 3
?? ?? ?
Page # 9
(v) .....
! 6
x
! 4
x
! 2
x
1 x cos
642
??? ? ?
(vi) tan x = ......
15
x 2
3
x
x
5 3
? ? ?
(vii) for |x| < 1, n ? R (1 + x)
n
= 1 + nx +
2 . 1
) 1 n ( n ?
x
2
+
3. 2. 1
) 2 n )( 1 n ( n ? ?
x
3
+ ............ ?
5. Limits of form 1
?
, 0
0
, ?
0
Also for (1)
?
type of  problems we can use following rules.
0 x
lim
?
(1 + x)
1/x
= e,
a x
lim
?
[f(x)]
g(x)
,
where f(x) ? 1   ;    g(x) ?? ??  as x ? a  =
)x ( g ] 1 )x ( f [ lim
a x
e
?
?
6. Sandwich Theorem or Squeeze Play Theorem:
If  f(x) ? g(x) ? h(x) ? x   &
a x
Limit
?
f(x) =
?
=
a x
Limit
?
h(x) then
a x
Limit
?
g(x) =
?
.
Page # 9
METHOD OF DIFFERENTIATION
1. Differentiation of some elementary functions
1.
dx
d
(x
n
) = nx
n – 1
2.
dx
d
(a
x
) = a
x
?n a
3.
dx
d
( ?n |x|) =
x
1
4. ?
dx
d
(log
a
x) =
a n x
1
?
5.
dx
d
(sin x) = cos x 6.
dx
d
(cos x) = – sin x
7.
dx
d
(sec x) = sec x  tan x 8.
dx
d
(cosec x) = – cosec x cot x
9.
dx
d
(tan x) = sec
2
x 10.
dx
d
(cot x) = – cosec
2
x
Page # 10
2.
1.
dx
d
(f ± g) = f ?(x) ± g ?(x) 2.
dx
d
(k f(x)) = k
dx
d
f(x)
3.
dx
d
(f(x) . g(x)) = f(x) g ?(x) + g(x) f ?(x)
4.
dx
d

?
?
?
?
?
?
?
?
) x ( g
) x ( f
=
) x ( g
)x ( g )x ( f ) x ( f )x ( g
2
? ? ?
5.
dx
d
(f(g(x))) =  f ?(g(x)) g ?(x)
Derivative Of Inverse Trigonometric Functions.
dx
x sin d
1 –
=
2
x 1
1
?
,
dx
x cos d
1 –
= –
2
x 1
1
?
, for – 1 < x < 1.
dx
x tan d
1 –
=
2
x 1
1
?
,
dx
x cot d
1 –
= –
2
x 1
1
?
(x ? R)
dx
x sec d
1 –
=
1 x | x |
1
2
?
,
dx
x ec cos d
1 –
= –
1 x | x |
1
2
?
, for x ? (– ?, – 1) ? (1, ?)
3. Differentiation using substitution
Following substitutions are normally used to simplify these expression.
(i)
2 2
a x ?
by substituting    x = a tan ?, where   –
2
?
< ? ? ? ?
2
?
(ii)
2 2
x a ?
by substituting    x = a sin ?, where   –
2
?
? ? ? ? ?
2
?
(iii)
2 2
a x ?
by substituting    x = a sec ?, where ? ? ? ?[0, ?], ? ? ?
2
?
(iv)
x a
a x
?
?
by substituting      x = a cos ?, where ? ? ? ?(0, ?].
Page # 11
4. Parametric Differentiation
If y = f( ?) & x = g( ?) where ? ?is a parameter, then
?
?
?
d / x d
d / y d
dx
dy
.
5. Derivative of one function with respect to another
Let y = f(x); z = g(x) then
) x ( ' g
) x ( ' f
x d / z d
xd / yd
z d
y d
? ? .
6. If F(x) =
)x( w )x( v )x( u
)x( n )x( m )x( l
)x( h )x( g )x( f
, where f, g, h, l, m, n, u, v, w are differentiable
functions of x then  F

? ?(x) =
)x ( w )x ( v )x ( u
)x ( n )x ( m )x ( l
) x (' h ) x (' g ) x (' f
+
)x ( w )x ( v )x ( u
)x ( ' n )x ( ' m )x ( ' l
)x ( h )x ( g )x ( f
+
) x (' w ) x (' v ) x (' u
)x ( n )x ( m ) x ( l
)x ( h )x ( g ) x ( f
```

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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