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JEE Class main Answered

Solve
Asked by sarveshvibrantacademy | 28 Apr, 2019, 11:17: PM
Expert Answer
Consider the limit
limit as straight x rightwards arrow 1 of fraction numerator open parentheses 1 minus straight x close parentheses open parentheses 1 minus straight x squared close parentheses open parentheses 1 minus straight x cubed close parentheses.... open parentheses 1 minus straight x to the power of 2 straight n end exponent close parentheses over denominator open parentheses open parentheses 1 minus straight x close parentheses open parentheses 1 minus straight x squared close parentheses open parentheses 1 minus straight x cubed close parentheses.... open parentheses 1 minus straight x to the power of straight n close parentheses close parentheses squared end fraction
equals limit as straight x rightwards arrow 1 of fraction numerator begin display style fraction numerator begin display style open parentheses 1 minus straight x close parentheses open parentheses 1 minus straight x squared close parentheses open parentheses 1 minus straight x cubed close parentheses.... open parentheses 1 minus straight x to the power of 2 straight n end exponent close parentheses end style over denominator begin display style open parentheses 1 minus straight x close parentheses to the power of 2 straight n end exponent end style end fraction end style over denominator begin display style fraction numerator begin display style open parentheses open parentheses 1 minus straight x close parentheses open parentheses 1 minus straight x squared close parentheses open parentheses 1 minus straight x cubed close parentheses.... open parentheses 1 minus straight x to the power of straight n close parentheses close parentheses squared end style over denominator begin display style open parentheses 1 minus straight x close parentheses to the power of 2 straight n end exponent end style end fraction end style end fraction
equals limit as straight x rightwards arrow 1 of fraction numerator begin display style fraction numerator 1 minus straight x over denominator 1 minus straight x end fraction cross times fraction numerator 1 minus straight x squared over denominator 1 minus straight x end fraction cross times fraction numerator begin display style 1 minus straight x cubed end style over denominator begin display style 1 minus straight x end style end fraction cross times... cross times fraction numerator begin display style 1 minus straight x to the power of 2 straight n end exponent end style over denominator begin display style 1 minus straight x end style end fraction end style over denominator begin display style open parentheses fraction numerator begin display style 1 minus straight x end style over denominator begin display style 1 minus straight x end style end fraction cross times fraction numerator begin display style 1 minus straight x squared end style over denominator begin display style 1 minus straight x end style end fraction cross times fraction numerator begin display style 1 minus straight x cubed end style over denominator begin display style 1 minus straight x end style end fraction cross times... cross times fraction numerator begin display style 1 minus straight x to the power of straight n end style over denominator begin display style 1 minus straight x end style end fraction close parentheses squared end style end fraction
Applying space straight L apostrophe Hospital space rule
equals limit as straight x rightwards arrow 1 of fraction numerator begin display style fraction numerator negative 1 over denominator negative 1 end fraction cross times fraction numerator negative 2 straight x over denominator negative 1 end fraction cross times fraction numerator negative 3 straight x squared over denominator negative 1 end fraction cross times... cross times fraction numerator negative 2 nx to the power of 2 straight n minus 1 end exponent over denominator negative 1 end fraction end style over denominator begin display style open parentheses fraction numerator negative 1 over denominator negative 1 end fraction cross times fraction numerator negative 2 straight x over denominator negative 1 end fraction cross times fraction numerator negative 3 straight x squared over denominator negative 1 end fraction cross times... cross times fraction numerator negative nx to the power of straight n minus 1 end exponent over denominator negative 1 end fraction close parentheses squared end style end fraction
equals fraction numerator 1 cross times 2 cross times 3 cross times... cross times 2 straight n over denominator begin display style open parentheses 1 cross times 2 cross times 3 cross times... cross times straight n close parentheses squared end style end fraction
equals fraction numerator 2 straight n factorial over denominator straight n factorial cross times straight n factorial end fraction

equals to the power of 2 straight n end exponent straight C subscript straight n
Coeff space of space straight x to the power of straight n space in space open parentheses 1 plus straight x close parentheses to the power of 2 straight n end exponent
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