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Please solve without using L'hopital rule
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Asked by g_archanasharma | 04 Mar, 2019, 11:35: AM
answered-by-expert Expert Answer
begin mathsize 12px style fraction numerator tan space x space minus space sin space x over denominator sin cubed x end fraction space equals space fraction numerator begin display style fraction numerator 1 over denominator cos begin display style x end style end fraction space minus space 1 end style over denominator begin display style sin squared x end style end fraction space space left curly bracket space N u m e r a t o r space a n d space d e n o m i n a t o r space a r e space d i v i d e d space b y space sin space x right curly bracket equals space fraction numerator 1 space minus space cos x over denominator cos begin display style x end style begin display style space end style begin display style sin squared end style begin display style x end style end fraction space equals space fraction numerator 2 space sin squared begin display style x over 2 end style over denominator cos begin display style x end style begin display style space end style begin display style sin squared end style begin display style x end style end fraction space equals space fraction numerator begin display style fraction numerator 2 space sin squared x over 2 over denominator begin display style open parentheses x over 2 close parentheses squared end style end fraction end style over denominator cos begin display style x end style begin display style space end style begin display style fraction numerator begin display style sin squared x end style over denominator open parentheses x over 2 close parentheses squared end fraction end style end fraction space space equals space fraction numerator fraction numerator 2 space sin squared x over 2 over denominator begin display style open parentheses x over 2 close parentheses squared end style end fraction over denominator cos begin display style space end style begin display style x end style begin display style space end style begin display style fraction numerator begin display style sin squared x end style over denominator x squared end fraction end style space begin display style 4 end style end fraction h e n c e space limit as x space rightwards arrow 0 of fraction numerator begin display style tan space x space minus space sin space x end style over denominator begin display style sin cubed x end style end fraction space equals space fraction numerator begin display style limit as x space rightwards arrow 0 of fraction numerator begin display style 2 space sin squared fraction numerator begin display style x end style over denominator begin display style 2 end style end fraction end style over denominator begin display style open parentheses x over 2 close parentheses squared end style end fraction end style over denominator begin display style limit as x space rightwards arrow 0 of end style cos begin display style space end style begin display style x end style begin display style space space space space limit as x space rightwards arrow 0 of end style begin display style fraction numerator begin display style sin squared x end style over denominator x squared end fraction end style space begin display style 4 end style end fraction space equals space 2 over 4 space equals space 1 half end style
Answered by Thiyagarajan K | 04 Mar, 2019, 14:08: PM

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