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Question Number 212137 by MrGaster last updated on 03/Oct/24

$$$$$$\underset{n\to \mathrm{\infty }}{\lim }(\sqrt[n]{n}-1)\sqrt{n}=?$$$$$$

Answered by Frix last updated on 03/Oct/24

$$\underset{n\to \mathrm{\infty }}{\lim }(n^{\frac{1}{n}+\frac{1}{2}}-n^{\frac{1}{2}})=\underset{t\to 0^{+}}{\lim }\frac{1-t^t}{t^{t+\frac{1}{2}}}\underset{\left[\mathrm{\&}\mathrm{transform}\right]}{\stackrel{\left[{\mathrm{l}}^{\prime }\mathrm{H}\widehat{\mathrm{o}pital}\right]}{=}}$$$$=-2\underset{t\to 0^{+}}{\lim }\frac{t^{\frac{1}{2}}(1+\ln t)}{1+2t(1+\ln t)}\stackrel{(\ast )}{=}\frac{0}{1}=0$$$$$$$$(\ast )\underset{t\to 0}{\lim }{t}^r\ln t=0\mathrm{\forall }r>0$$

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