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  4. displaystyle lim n arrow ∞ n2(x1 / n-x1 / n+1), x>0 is equal to

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Q. $\displaystyle\lim _{n \rightarrow \infty} n^{2}\left(x^{1 / n}-x^{1 / n+1}\right), x>0$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{n \rightarrow \infty} n^{2}\left(x^{1 / n}-x^{\frac{1}{n+1}}\right)$
$=\displaystyle\lim _{n \rightarrow \infty} n^{2} \cdot x^{\frac{1}{n+1}}\left(x^{\frac{1}{n}-\frac{1}{n+1}}-1\right)$
$=\displaystyle\lim _{n \rightarrow \infty} x^{\frac{1}{n+1}}\left(\frac{1}{x^{n(n+1)}}-1\right) n^{2}$
$=\displaystyle\lim _{n \rightarrow \infty} x^{\frac{1}{n+1}} \cdot \frac{x^{\frac{1}{n(n+1)}}-1}{\frac{1}{n(n+1)}} \cdot \frac{n^{2}}{n(n+1)}$
$=1 \cdot \ln x \cdot 1=\ln x$