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Solve the 3rd question
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Asked by g_archanasharma | 09 Apr, 2019, 21:09: PM
answered-by-expert Expert Answer
G i v e n space f u n c t i o n space i s space f left parenthesis x right parenthesis equals sin to the power of negative 1 end exponent left parenthesis cos space x right parenthesis space for all x element of R N o w comma space L H L equals limit as x rightwards arrow 0 to the power of minus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of sin to the power of negative 1 end exponent left parenthesis cos space x right parenthesis equals sin to the power of negative 1 end exponent left parenthesis 0 right parenthesis equals straight pi space and space RHL equals limit as straight x rightwards arrow 0 to the power of plus of straight f left parenthesis straight x right parenthesis equals limit as straight x rightwards arrow 0 of sin to the power of negative 1 end exponent left parenthesis cos space straight x right parenthesis equals sin to the power of negative 1 end exponent left parenthesis 0 right parenthesis equals straight pi space equals space straight f left parenthesis 0 right parenthesis therefore LHL equals RHL equals straight f left parenthesis 0 right parenthesis Thus comma space straight f left parenthesis straight x right parenthesis space is space continuous space at space straight x equals 0 Now comma space to space check space differentiability space of space straight f left parenthesis straight x right parenthesis space at space straight x equals 0 straight f apostrophe left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator square root of 1 minus cos squared straight x end root end fraction left parenthesis negative sin space straight x right parenthesis equals negative fraction numerator sin space straight x over denominator vertical line sin space straight x vertical line end fraction Now comma space limit as straight x rightwards arrow 0 to the power of minus of straight f apostrophe left parenthesis straight x right parenthesis equals 1 space and space limit as straight x rightwards arrow 0 to the power of plus of straight f apostrophe left parenthesis straight x right parenthesis equals negative 1 therefore limit as straight x rightwards arrow 0 to the power of minus of straight f apostrophe left parenthesis straight x right parenthesis not equal to limit as straight x rightwards arrow 0 to the power of plus of straight f apostrophe left parenthesis straight x right parenthesis Hence comma space straight f left parenthesis straight x right parenthesis space is space not space differentiable space at space straight x equals 0.
Answered by Renu Varma | 10 Apr, 2019, 10:42: AM

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