सिद्ध कीजिए कि फलनों का अनुक्रम {fn} जहाँ
अनुक्रम $\{f_n\}$ को परिभाषित किया गया है: $$ f_n(x) = \frac{n}{x+n}. $$ हम इसकी सीमा __________ ______ ________ _________ __________ ______ ________ ________ ______ ________.
_________ ______ _________ _________ _________ _________.
__________ _______ ______ _________ ______ _____ _______ _______ ________.
______ _________ ________ ___ ______ ________ ___ ___.
_______ ________ _____ ________ ___ ________.
_________ ________ _______ ___ ______ __________ __________ _____ ___ ____ ________.
_________ ___ ____ ___ __________ ___ ________ __________ __________ ______ _________.
___.
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Test the following series for convergence.
(i)
(ii)
Disprove the statement:
by providing a suitable counter-example.
Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:
i) Every infinite set is an open set.
ii) The negation of p^~qis p→q.
iii) -1 is a limit point of the interval ]-2,1].
iv) The necessary condition for a function f to be integrable is that it is continuous.
The function f: R→R defined by f(x) = |x-2|+|3-x is differentiable at x = 5.
सिद्ध कीजिए कि ex cos2x = 2 के किन्हीं दो वास्तविक मूलों के बीच ex sin2x = 1 का कम से कम एक वास्तविक मूल होता है।
सिद्ध कीजिए :
Prove that a function f : S → S (where S is a finite non-empty set) is injective if it is surjective.
Prove that the sequence (an)n∈N, where an = 32 / x2 + 22 , is Cauchy.