JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > Important Formulas: Limits & Derivatives
Important Limits & Derivatives Formulas for JEE and NEET
| Download, print and study this document offline |
Please wait while the PDF view is loading
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
Read More
|
209 videos|443 docs|143 tests
|
About this Document
|
4.74/5 Rating |
|
Dec 22, 2024 Last updated |
Document Description: Important Formulas: Limits & Derivatives for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation.
The notes and questions for Important Formulas: Limits & Derivatives have been prepared according to the JEE exam syllabus. Information about Important Formulas: Limits & Derivatives covers topics
like and Important Formulas: Limits & Derivatives Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Important Formulas: Limits & Derivatives.
Introduction of Important Formulas: Limits & Derivatives in English is available as part of our Mathematics (Maths) for JEE Main & Advanced
for JEE & Important Formulas: Limits & Derivatives in Hindi for Mathematics (Maths) for JEE Main & Advanced course.
Download more important topics related with notes, lectures and mock test series for JEE
Exam by signing up for free. JEE: Important Limits & Derivatives Formulas for JEE and NEET
Description
Full syllabus notes, lecture & questions for Important Limits & Derivatives Formulas for JEE and NEET - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download
Information about Important Formulas: Limits & Derivatives
In this doc you can find the meaning of Important Formulas: Limits & Derivatives defined & explained in the simplest way possible. Besides explaining types of
Important Formulas: Limits & Derivatives theory, EduRev gives you an ample number of questions to practice Important Formulas: Limits & Derivatives tests, examples and also practice JEE
tests
|
209 videos|443 docs|143 tests
|
Download as PDF
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
video lectures
,Semester Notes
,Free
,shortcuts and tricks
,Important questions
,MCQs
,ppt
,Sample Paper
,Important Limits & Derivatives Formulas for JEE and NEET
,Summary
,Objective type Questions
,Previous Year Questions with Solutions
,Exam
,Important Limits & Derivatives Formulas for JEE and NEET
,mock tests for examination
,Viva Questions
,Extra Questions
,past year papers
,practice quizzes
,study material
,Important Limits & Derivatives Formulas for JEE and NEET
;
Additional Information about Important Formulas: Limits & Derivatives for JEE Preparation
Important Formulas: Limits & Derivatives Free PDF Download
The Important Formulas: Limits & Derivatives is an invaluable resource that delves deep into the core of the JEE exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Important Formulas: Limits & Derivatives now and kickstart your journey towards success in the JEE exam.
Importance of Important Formulas: Limits & Derivatives
The importance of Important Formulas: Limits & Derivatives cannot be overstated, especially for JEE aspirants.
This document holds the key to success in the JEE exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Important Formulas: Limits & Derivatives Notes
Important Formulas: Limits & Derivatives Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Important Formulas: Limits & Derivatives.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Important Formulas: Limits & Derivatives Notes on EduRev are your ultimate resource for success.
Important Formulas: Limits & Derivatives JEE Questions
The "Important Formulas: Limits & Derivatives JEE Questions" guide is a valuable resource for all aspiring students preparing for the
JEE exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Important Formulas: Limits & Derivatives on the App
Students of JEE can study Important Formulas: Limits & Derivatives alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Important Formulas: Limits & Derivatives,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Important Formulas: Limits & Derivatives is prepared as per the latest JEE syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup