Question
Are the following statements true or false? Give reasons for tour answers.
a) -2 is a limit point of the interval 1-3,2].
b) The series is divergent.
The function, f(x)=sin² x is uniformly continuous in the interval [0, π).
d) Every continuous function is differentiable.
e) The function f defined on R by
is irrational Is integrable in the interval [2,3].
Answer :
Word Count : 567
Here are the answers along with explanations: --- ### (a) -2 is a limit point of the interval \( (1-3,2] \). False. A limit point of a set \( S \) is a point where every neighborhood contains at least one point of \( S \) different from itself. The given interval appears to be a typo—assuming you meant \( (1,3,2] \), it is not a well-defined interval. If you meant \( (1,3] \), then -2 is clearly not a limit point, since no sequence from \( (1,3] \) can approach -2. --- ### _____ __________ __________ _____ ____ ________ _______ _____ ______ ___.
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Here are the answers along with explanations: --- ### (a) -2 is a limit point of the interval \( (1-3,2] \). False. A limit point of a set \( S \) is a point where every neighborhood contains at least one point of \( S \) different from itself. The given interval appears to be a typo—assuming you meant \( (1,3,2] \), it is not a well-defined interval. If you meant \( (1,3] \), then -2 is clearly not a limit point, since no sequence from \( (1,3] \) can approach -2. --- ### _____ __________ __________ _____ ____ ________ _______ _____ ______ ___.
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Related Questions
निम्नलिखित श्रेणी के अभिसरण की जाँच कीजिए:
i)
ii)
दिखाइए कि अनुक्रम (an), एकदिष्ट है, जहाँ उत्तर की पुष्टि कीजिए। । क्या (an) कॉशी अनुक्रम है? अपने
Give an example of a series ∑ such that ∑
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निरपेक्ष और सप्रतिबंध अभिसरण के लिए श्रेणी
की जाँच कीजिए।
सिद्ध कीजिए कि अनुक्रम अभिसारी। है जहाँ {an} परिबद्ध अनुक्रम है।
Prove that the function defined by
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, using the sequential definition of continuity.
Prove that the sequence is convergent where (
n) is a bounded sequence.
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