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Asked by g_archanasharma | 02 Mar, 2019, 19:38: PM
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begin mathsize 16px style limit as n rightwards arrow infinity of C presuperscript n subscript x open parentheses m over n close parentheses to the power of x open parentheses 1 minus m over n close parentheses to the power of n minus x end exponent equals space limit as n rightwards arrow infinity of fraction numerator n open parentheses n minus 1 close parentheses open parentheses n minus 2 close parentheses.......... left parenthesis n minus x plus 1 right parenthesis over denominator x factorial end fraction open parentheses m over n close parentheses to the power of x open parentheses 1 minus m over n close parentheses to the power of n over open parentheses 1 minus m over n close parentheses to the power of x equals space limit as n rightwards arrow infinity of fraction numerator n to the power of x space 1 open parentheses 1 minus begin display style 1 over n end style close parentheses open parentheses 1 minus begin display style 2 over n end style close parentheses........... open parentheses 1 minus begin display style fraction numerator x minus 1 over denominator n end fraction end style close parentheses over denominator x factorial end fraction m to the power of x over n to the power of x open curly brackets open parentheses 1 plus fraction numerator 1 over denominator begin display style bevelled fraction numerator negative n over denominator m end fraction end style end fraction close parentheses to the power of begin display style bevelled fraction numerator negative n over denominator m end fraction end style end exponent close curly brackets to the power of negative m end exponent over open parentheses 1 minus m over n close parentheses to the power of x end style
 
when n→∞ ,  (1/n) terms are zero.  Let u = -n/m 
 
Hence the limit becomes    begin mathsize 16px style fraction numerator m to the power of x over denominator x factorial end fraction limit as u rightwards arrow infinity of open curly brackets open parentheses 1 plus 1 over u close parentheses to the power of u close curly brackets to the power of negative m end exponent space space equals space fraction numerator m to the power of x over denominator x factorial end fraction e to the power of negative m end exponent end style
Answered by Thiyagarajan K | 03 Mar, 2019, 14:24: PM

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