Engineering
Mathematics
,Published By:
Physics Wallah
ISBN: 978-93-94342-39-2
Mobile App: Physics Wallah (Available on Play Store)
Website: www.pw.live
Email:
Rights
All rights will be reserved by Publisher. No part of this book may be used or reproduced in any manner
whatsoever without the written permission from author or publisher.
In the interest of student's community:
Circulation of soft copy of Book(s) in PDF or other equivalent format(s) through any social media channels,
emails, etc. or any other channels through mobiles, laptops or desktop is a criminal offence. Anybody
circulating, downloading, storing, soft copy of the book on his device(s) is in breach of Copyright Act. Further
Photocopying of this book or any of its material is also illegal. Do not download or forward in case you come
across any such soft copy material.
Disclaimer
A team of PW experts and faculties with an understanding of the subject has worked hard for the books.
While the author and publisher have used their best efforts in preparing these books. The content has been
checked for accuracy. As the book is intended for educational purposes, the author shall not be responsible for
any errors contained in the book.
The publication is designed to provide accurate and authoritative information with regard to the subject matter
covered.
This book and the individual contribution contained in it are protected under copyright by the publisher.
(This Module shall only be Used for Educational Purpose.)
, Design Against Static Load
INDEX
1. Basic Calculus ...................................................................................................................... 12.1 – 12.14
2. Differential Equations........................................................................................................... 12.15 – 12.24
3. Vector Calculus .................................................................................................................... 12.25 – 12.30
4. Linear Algebra ...................................................................................................................... 12.31 – 12.36
5. Probability and Statistics ...................................................................................................... 12.37 – 12.46
6. Numerical Methods .............................................................................................................. 12.47 – 12.50
7. Complex Calculus ................................................................................................................. 12.51 – 12.54
GATE-O-PEDIA MECHANICAL ENGINEERING
, Design Against Static Load
1
1.1 Introduction
BASIC CALCULUS
1.1.1 Limits, Continuity and Differentiability
a. As x tends to a (x → a) x is moving towards a
A value l is said to be limit of a function f(x) at x → a if f(x) → l as x → a.
It is mathematically defined as
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 = 𝑙𝑖𝑚− 𝑓(𝑥) = 𝑙𝑖𝑚+ 𝑓(𝑥)
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
A function f(x) is said to be continuous at x = a if
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 = 𝑓(𝑎) = 𝑓(𝑥)|𝑥=𝑎
𝑥→𝑎
Note:
For 𝑙𝑖𝑚 𝑓(𝑥) to exist, the function need not be continuous at x = a.
𝑥→𝑎
But for f(x) to be continuous at x = a, 𝑙𝑖𝑚 𝑓(𝑎) should exist.
𝑥→𝑎
Fig. 1.1
GATE WALLAH MECHANICAL HANDBOOK 12.1
Mathematics
,Published By:
Physics Wallah
ISBN: 978-93-94342-39-2
Mobile App: Physics Wallah (Available on Play Store)
Website: www.pw.live
Email:
Rights
All rights will be reserved by Publisher. No part of this book may be used or reproduced in any manner
whatsoever without the written permission from author or publisher.
In the interest of student's community:
Circulation of soft copy of Book(s) in PDF or other equivalent format(s) through any social media channels,
emails, etc. or any other channels through mobiles, laptops or desktop is a criminal offence. Anybody
circulating, downloading, storing, soft copy of the book on his device(s) is in breach of Copyright Act. Further
Photocopying of this book or any of its material is also illegal. Do not download or forward in case you come
across any such soft copy material.
Disclaimer
A team of PW experts and faculties with an understanding of the subject has worked hard for the books.
While the author and publisher have used their best efforts in preparing these books. The content has been
checked for accuracy. As the book is intended for educational purposes, the author shall not be responsible for
any errors contained in the book.
The publication is designed to provide accurate and authoritative information with regard to the subject matter
covered.
This book and the individual contribution contained in it are protected under copyright by the publisher.
(This Module shall only be Used for Educational Purpose.)
, Design Against Static Load
INDEX
1. Basic Calculus ...................................................................................................................... 12.1 – 12.14
2. Differential Equations........................................................................................................... 12.15 – 12.24
3. Vector Calculus .................................................................................................................... 12.25 – 12.30
4. Linear Algebra ...................................................................................................................... 12.31 – 12.36
5. Probability and Statistics ...................................................................................................... 12.37 – 12.46
6. Numerical Methods .............................................................................................................. 12.47 – 12.50
7. Complex Calculus ................................................................................................................. 12.51 – 12.54
GATE-O-PEDIA MECHANICAL ENGINEERING
, Design Against Static Load
1
1.1 Introduction
BASIC CALCULUS
1.1.1 Limits, Continuity and Differentiability
a. As x tends to a (x → a) x is moving towards a
A value l is said to be limit of a function f(x) at x → a if f(x) → l as x → a.
It is mathematically defined as
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 = 𝑙𝑖𝑚− 𝑓(𝑥) = 𝑙𝑖𝑚+ 𝑓(𝑥)
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
A function f(x) is said to be continuous at x = a if
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 = 𝑓(𝑎) = 𝑓(𝑥)|𝑥=𝑎
𝑥→𝑎
Note:
For 𝑙𝑖𝑚 𝑓(𝑥) to exist, the function need not be continuous at x = a.
𝑥→𝑎
But for f(x) to be continuous at x = a, 𝑙𝑖𝑚 𝑓(𝑎) should exist.
𝑥→𝑎
Fig. 1.1
GATE WALLAH MECHANICAL HANDBOOK 12.1
