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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 ?? ??? ??? 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1. What is the definition of a limit in calculus? 
2. How do you find the limit of a function using algebraic manipulation? 
3. What is the difference between a onesided limit and a twosided limit? 
4. How do you calculate the derivative of a function using the limit definition? 
5. What is the relationship between limits and derivatives in calculus? 
209 videos443 docs143 tests


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