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Continuity And Differentiability

Question

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Is the function defined by f(x) = x² –sin x + 5 continuous at x =?

Solution

straight f left parenthesis straight x right parenthesis equals straight x squared minus space sinx plus 5 Lt with straight x rightwards arrow straight pi below space straight f left parenthesis straight x right parenthesis equals stack space Lt space with straight x rightwards arrow straight pi below left parenthesis straight x squared minus space sinx space plus 5 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals straight pi squared minus space sinπ plus 5 equals straight pi squared minus 0 plus 5 equals straight pi squared plus 5 Also space straight f left parenthesis straight pi right parenthesis space equals straight pi squared minus space sinπ plus 5 equals straight pi squared minus 0 plus 5 equals straight pi squared plus 5 therefore space Lt with straight x rightwards arrow straight pi below space straight f left parenthesis straight pi right parenthesis equals straight f left parenthesis straight pi right parenthesis therefore space space straight f space is space continous space at space straight x equals straight pi.

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