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\( \lim _{x \rightarrow 0}\left(\frac{1^{x}+2^{x}+3^{x}+\ldots .+n^{x}}{n}\right)^{n / x} \) |
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Answer» \(\underset{x\rightarrow0}{lim}\) \((\frac{1^x+2^x+3^x+...+n^x}n)^{n/x}\) (1∞ type) = Exp \((\underset{x\rightarrow0}{lim}\) \(n/x\times(\frac{1^x+2^x+3^x+...+n^x}n-1))\) = Exp \((\underset{x\rightarrow0}{lim}\) \(\frac{1^x+2^x+3^x+...+n^x-n}x)\) = Exp \((\underset{x\rightarrow0}{lim}\) \(\frac{1^xlog\,1+2^xlog\,2+3^xlog\,3+...+n^xlog\,n}1)\) (By using D.L.H. Rule) = Exp \(\{\underset{x\rightarrow0}{lim}\) 2x log 2 + 3x log 3 + .... + nx log n} (∵ log 1 = 0) = Exp {log 2 + log 3 + ... + log n} = Exp {log (2 x 3 x 4 x ... x n)} (∵ log A + log B = log AB) = 1 x 2 x 3 x 4 x ... x n (∵ elog x = x) = n! Hence, \(\underset{x\rightarrow0}{lim}\) \((\frac{1^x+2^x+3^x+...+n^x}n)^{n/x}\) = n! |
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