Asked by Topperlearning User | 28th Jul, 2014, 10:53: AM

Expert Answer:

limit as x rightwards arrow pi over 2 of open parentheses fraction numerator 1 plus c o t space x over denominator 1 plus cos space x end fraction close parentheses to the power of fraction numerator 1 over denominator cos space x end fraction end exponent
equals limit as x rightwards arrow pi over 2 of open parentheses 1 plus c o t space x close parentheses to the power of fraction numerator 1 over denominator cos space x space end fraction end exponent obelus divided by limit as x rightwards arrow pi over 2 of open parentheses 1 plus cos space x close parentheses to the power of fraction numerator 1 over denominator cos space x end fraction end exponent
L e t space cos space x equals z. space I f space x rightwards arrow pi over 2 comma space t h e n space z rightwards arrow 0
equals limit as x rightwards arrow pi over 2 of open square brackets open parentheses 1 plus c o t space x close parentheses to the power of fraction numerator 1 over denominator begin display style fraction numerator cos space x over denominator sin space x end fraction end style space end fraction. end exponent close square brackets to the power of limit as x rightwards arrow pi over 2 of fraction numerator 1 over denominator sin space x end fraction end exponent obelus divided by limit as z rightwards arrow 0 of open parentheses 1 plus z close parentheses to the power of 1 over z end exponent
L e t space p equals c o t space x. space I f space x rightwards arrow pi over 2 comma space t h e n space p rightwards arrow 0
equals limit as p rightwards arrow 0 of open square brackets open parentheses 1 plus p close parentheses to the power of fraction numerator 1 over denominator begin display style p end style space end fraction end exponent close square brackets obelus divided by limit as z rightwards arrow 0 of open parentheses 1 plus z close parentheses to the power of 1 over z end exponent
equals e obelus divided by e equals 1

Answered by  | 28th Jul, 2014, 12:53: PM