CBSE Class 11-science Answered

Asked by Topperlearning User | 28 Jul, 2014, 10:02: AM

Expert Answer

$limit as x rightwards arrow 0 of open parentheses fraction numerator tan space x over denominator x end fraction close parentheses to the power of 1 over x squared end exponent equals e to the power of limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction end exponent space space open square brackets stack lim space with x rightwards arrow 0 below f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent space equals e to the power of limit as x rightwards arrow 0 of g left parenthesis x right parenthesis. log open square brackets f left parenthesis x right parenthesis close square brackets end exponent close square brackets equals e to the power of limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction end exponent L e t space A equals to the power of limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction space space space open square brackets 0 over 0 space f o r m close square brackets end exponent limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction space left square bracket L apostrophe space H o s p i t a l apostrophe s space R u l e right square bracket equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator up diagonal strike x over denominator tan space x end fraction end style. begin display style fraction numerator x space s e c squared space x minus tan space x over denominator x to the power of up diagonal strike 2 end exponent end fraction end style over denominator 2 x end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator tan space x end fraction end style. begin display style fraction numerator x space s e c squared space x minus tan space x over denominator x end fraction end style over denominator 2 x end fraction space equals limit as x rightwards arrow 0 of fraction numerator begin display style x space s e c squared end style x begin display style minus tan space x end style over denominator 2 x squared tan space x end fraction space open square brackets 0 over 0 space f o r m close square brackets equals limit as x rightwards arrow 0 of fraction numerator begin display style space s e c squared end style x begin display style plus x.2 s e c squared x. tan space x end style minus s e c squared x over denominator 4 x. tan space x plus 2 x squared. s e c squared space x end fraction left square bracket L apostrophe space H o s p i t a l apostrophe s space r u l e right square bracket equals limit as x rightwards arrow 0 of fraction numerator begin display style 2 x. s e c squared x. tan space x end style over denominator 2 x left parenthesis tan space x plus x. s e c squared space x right parenthesis end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style s e c squared x. tan space x end style over denominator left parenthesis tan space x plus x. s e c squared space x right parenthesis end fraction space space open square brackets 0 over 0 space f o r m close square brackets equals limit as x rightwards arrow 0 of fraction numerator begin display style s e c to the power of 4 x plus 2 s e c squared x. tan squared x end style over denominator s e c squared x plus s e c squared space x plus 2 x. s e c squared x. tan space x end fraction open square brackets L apostrophe H o s p i t a l apostrophe s space r u l e close square brackets equals limit as x rightwards arrow 0 of fraction numerator begin display style s e c squared x left parenthesis 1 plus 2. tan squared x right parenthesis end style over denominator s e c squared space x left parenthesis 2 plus 2 x. tan space x right parenthesis end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style left parenthesis 1 plus 2. tan squared x right parenthesis end style over denominator left parenthesis 2 plus 2 x. tan space x right parenthesis end fraction equals 1 half therefore A equals 1 half therefore limit as x rightwards arrow 0 of open parentheses fraction numerator tan space x over denominator x end fraction close parentheses to the power of 1 over x squared end exponent equals e to the power of 1 half end exponent$

Answered by | 28 Jul, 2014, 12:02: PM

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CBSE Class 11-science Answered

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