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長文書きたいときに使う.

京大1973年理系問題4

\displaystyle\lim_{n\to\infty}n\Biggl[\frac{(\sqrt{n(n+1)}-n)^3}n-\frac{(\sqrt{n(n+1)}-(n+1))^3}{n+1}\Biggr]を求めよ.

(与式)
=\displaystyle\lim_{n\to\infty}\Biggl[\left(\sqrt{n(n+1)}-n\right)^3-\frac n{n+1}\left(\sqrt{n(n+1)}-(n+1)\right)^3\Biggr]
=\displaystyle\lim_{n\to\infty}\Biggl[\left(\frac n{\sqrt{n(n+1)}+n}\right)^3-\frac n{n+1}\left(-\frac{n+1}{\sqrt{n(n+1)}+(n+1)}\right)^3\Biggr]
=\displaystyle\left(\frac12\right)^3-1\cdot\left(-\frac12\right)^3
=\dfrac14.