Flashcards: Limits and Derivatives | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
 
 
LIMITS and Derivatives 
Let ?? = ?? (?? ) be a real valued function which is defined in the neighbourhood of the point at 
?? = ?? . A real number ?? is called limit of the function ?? when ?? ? ?? i.e., lim
?? ??? ??? (?? ) = ?? . If 
given ?? > 0, however small, there exists ?? (?? ) > 0 such that |?? (?? ) - ?? | < ?? whenever 
|?? - ?? | < ?? . The notion of continuity occurs in many application of calculus. We may 
intuitively think of continuous function as those functions whose graphs we can draw 
without lifting the pencil off the paper. 
  
Page 2


 
 
 
 
 
LIMITS and Derivatives 
Let ?? = ?? (?? ) be a real valued function which is defined in the neighbourhood of the point at 
?? = ?? . A real number ?? is called limit of the function ?? when ?? ? ?? i.e., lim
?? ??? ??? (?? ) = ?? . If 
given ?? > 0, however small, there exists ?? (?? ) > 0 such that |?? (?? ) - ?? | < ?? whenever 
|?? - ?? | < ?? . The notion of continuity occurs in many application of calculus. We may 
intuitively think of continuous function as those functions whose graphs we can draw 
without lifting the pencil off the paper. 
  
 
 
 
 
 
EXISTENCE OF LIMIT 
lim
?? ??? ??? (?? ) = ?? exists if 
lim
?? ??? +??? (?? ) = lim
?? ??? -??? (?? ) = finite and definite 
or lim
h?0
??? (?? + h) = lim
h?0
??? (?? - h) = finite and definite 
where lim
?? ??? +??? (?? ) is called right hand limit 
and lim
?? ??? -??? (?? ) is called left hand limit. 
  
Page 3


 
 
 
 
 
LIMITS and Derivatives 
Let ?? = ?? (?? ) be a real valued function which is defined in the neighbourhood of the point at 
?? = ?? . A real number ?? is called limit of the function ?? when ?? ? ?? i.e., lim
?? ??? ??? (?? ) = ?? . If 
given ?? > 0, however small, there exists ?? (?? ) > 0 such that |?? (?? ) - ?? | < ?? whenever 
|?? - ?? | < ?? . The notion of continuity occurs in many application of calculus. We may 
intuitively think of continuous function as those functions whose graphs we can draw 
without lifting the pencil off the paper. 
  
 
 
 
 
 
EXISTENCE OF LIMIT 
lim
?? ??? ??? (?? ) = ?? exists if 
lim
?? ??? +??? (?? ) = lim
?? ??? -??? (?? ) = finite and definite 
or lim
h?0
??? (?? + h) = lim
h?0
??? (?? - h) = finite and definite 
where lim
?? ??? +??? (?? ) is called right hand limit 
and lim
?? ??? -??? (?? ) is called left hand limit. 
  
 
 
 
 
 
EXISTENCE OF LIMIT 
Conclusion : 
lim
?? ??? ??? (?? ) exist if its left hand limit (LHL) at ?? = ?? , and right hand limit (RHL) at ?? = ?? both exist and 
are equal and finite. i.e., lim
?? ??? +??? (?? ) = lim
?? ??? -??? (?? ) = finite, otherwise limit does not exist. 
  
Page 4


 
 
 
 
 
LIMITS and Derivatives 
Let ?? = ?? (?? ) be a real valued function which is defined in the neighbourhood of the point at 
?? = ?? . A real number ?? is called limit of the function ?? when ?? ? ?? i.e., lim
?? ??? ??? (?? ) = ?? . If 
given ?? > 0, however small, there exists ?? (?? ) > 0 such that |?? (?? ) - ?? | < ?? whenever 
|?? - ?? | < ?? . The notion of continuity occurs in many application of calculus. We may 
intuitively think of continuous function as those functions whose graphs we can draw 
without lifting the pencil off the paper. 
  
 
 
 
 
 
EXISTENCE OF LIMIT 
lim
?? ??? ??? (?? ) = ?? exists if 
lim
?? ??? +??? (?? ) = lim
?? ??? -??? (?? ) = finite and definite 
or lim
h?0
??? (?? + h) = lim
h?0
??? (?? - h) = finite and definite 
where lim
?? ??? +??? (?? ) is called right hand limit 
and lim
?? ??? -??? (?? ) is called left hand limit. 
  
 
 
 
 
 
EXISTENCE OF LIMIT 
Conclusion : 
lim
?? ??? ??? (?? ) exist if its left hand limit (LHL) at ?? = ?? , and right hand limit (RHL) at ?? = ?? both exist and 
are equal and finite. i.e., lim
?? ??? +??? (?? ) = lim
?? ??? -??? (?? ) = finite, otherwise limit does not exist. 
  
 
 
 
Indeterminate Forms 
A functional form whose value can't be fixed is called indeterminate form. For Example, 
0
0
 is an 
indeterminate form as if, we take 
0
0
= ?? ? 0 = ?? × 0. In this case we have infinitely many finite values of 
?? for which ?? × 0 = 0. Thus ?? is not fixed. Hence [
0
0
] form is indeterminate form. Some indeterminate 
forms are 
8
8
, 0 × 8, 8 - 8, 0
0
, 8
0
, 1
8
 etc. 
  
Page 5


 
 
 
 
 
LIMITS and Derivatives 
Let ?? = ?? (?? ) be a real valued function which is defined in the neighbourhood of the point at 
?? = ?? . A real number ?? is called limit of the function ?? when ?? ? ?? i.e., lim
?? ??? ??? (?? ) = ?? . If 
given ?? > 0, however small, there exists ?? (?? ) > 0 such that |?? (?? ) - ?? | < ?? whenever 
|?? - ?? | < ?? . The notion of continuity occurs in many application of calculus. We may 
intuitively think of continuous function as those functions whose graphs we can draw 
without lifting the pencil off the paper. 
  
 
 
 
 
 
EXISTENCE OF LIMIT 
lim
?? ??? ??? (?? ) = ?? exists if 
lim
?? ??? +??? (?? ) = lim
?? ??? -??? (?? ) = finite and definite 
or lim
h?0
??? (?? + h) = lim
h?0
??? (?? - h) = finite and definite 
where lim
?? ??? +??? (?? ) is called right hand limit 
and lim
?? ??? -??? (?? ) is called left hand limit. 
  
 
 
 
 
 
EXISTENCE OF LIMIT 
Conclusion : 
lim
?? ??? ??? (?? ) exist if its left hand limit (LHL) at ?? = ?? , and right hand limit (RHL) at ?? = ?? both exist and 
are equal and finite. i.e., lim
?? ??? +??? (?? ) = lim
?? ??? -??? (?? ) = finite, otherwise limit does not exist. 
  
 
 
 
Indeterminate Forms 
A functional form whose value can't be fixed is called indeterminate form. For Example, 
0
0
 is an 
indeterminate form as if, we take 
0
0
= ?? ? 0 = ?? × 0. In this case we have infinitely many finite values of 
?? for which ?? × 0 = 0. Thus ?? is not fixed. Hence [
0
0
] form is indeterminate form. Some indeterminate 
forms are 
8
8
, 0 × 8, 8 - 8, 0
0
, 8
0
, 1
8
 etc. 
  
 
 
 
Algebra of Limits 
If lim
?? ??? ??? (?? ) and lim
?? ??? ??? (?? ) both exist then 
(i)  lim
?? ??? ?[?? (?? ) ± ?? (?? )] = lim
?? ??? ??? (?? ) ± lim
?? ??? ??? (?? ) 
(ii) lim
?? ??? ?[
?? (?? )
?? (?? )
] =
lim
?? ??? ??? (?? )
lim
?? ??? ??? (?? )
 ; (lim
?? ??? ??? (?? ) ? 0) 
(iii) lim
?? ??? ?[???? (?? )] = ?? lim
?? ??? ??? (?? )