Asked by Topperlearning User | 28th 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  | 28th Jul, 2014, 12:02: PM