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Limits And Derivatives

Question
CBSEENMA11013152
Wired Faculty App

Evaluate space space limit as straight x rightwards arrow straight x over 4 of fraction numerator cos space straight x space minus space sin space straight x over denominator straight x minus straight pi divided by 4 end fraction.

Solution

space space space space space space space space space space space limit as straight x rightwards arrow straight pi over 4 of fraction numerator cos space space straight x space minus space sin space straight x over denominator straight x space minus space straight pi divided by 4 end fraction
Put  space space straight x space equals straight pi over 4 plus straight h comma space straight x rightwards arrow straight pi over 4 rightwards double arrow straight h rightwards arrow 0              

                    

∴           space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow straight pi over 4 of fraction numerator cos space straight x space minus space sin space straight x over denominator straight x minus begin display style straight pi over 4 end style end fraction equals limit as straight h rightwards arrow 0 of fraction numerator cos open parentheses begin display style straight pi over 4 end style plus straight h close parentheses minus sin open parentheses begin display style straight pi over 4 end style plus straight h close parentheses over denominator begin display style straight pi over 4 end style plus straight h minus begin display style straight pi over 4 end style end fraction

        space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator open parentheses cos straight pi over 4. cos space straight h minus sin straight pi over 4. sin space straight h close parentheses minus open parentheses sin straight pi over 4. cos space straight h minus sin straight pi over 4. sin space straight h close parentheses over denominator straight h end fraction

        space space space space space space space space space space space space space space space equals space space space limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style cos space straight h space space minus begin display style fraction numerator 1 over denominator square root of 2 end fraction end style sin space straight h space minus space fraction numerator 1 over denominator square root of 2 end fraction cos space straight h space space minus fraction numerator 1 over denominator square root of 2 end fraction sin space straight h over denominator straight h end fraction

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           space space space space space space space space equals space space fraction numerator negative 2 over denominator square root of 2 end fraction straight x fraction numerator square root of 2 over denominator square root of 2 end fraction equals negative square root of 2
     

          

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