In the recent sudden quiz, there is a so-called bonus question (actually it is a boring 1-mark question): evaluate
![\displaystyle \lim_{n\rightarrow \infty}(\sqrt[8]{n^2+1}-\sqrt[4]{n+1})](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clim_%7Bn%5Crightarrow+%5Cinfty%7D%28%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D-%5Csqrt%5B4%5D%7Bn%2B1%7D%29&bg=ffffff&fg=000000&s=0 "\displaystyle \lim_{n\rightarrow \infty}(\sqrt[8]{n^2+1}-\sqrt[4]{n+1})")
Here is a suggested solution
![=\displaystyle \lim_{n\rightarrow \infty}\frac{\sqrt[4]{n^2+1}-\sqrt{n+1}}{\sqrt[8]{n^2+1}+\sqrt[4]{n+1}}](http://s0.wp.com/latex.php?latex=%3D%5Cdisplaystyle+%5Clim_%7Bn%5Crightarrow+%5Cinfty%7D%5Cfrac%7B%5Csqrt%5B4%5D%7Bn%5E2%2B1%7D-%5Csqrt%7Bn%2B1%7D%7D%7B%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%5B4%5D%7Bn%2B1%7D%7D&bg=ffffff&fg=000000&s=0 "=\displaystyle \lim_{n\rightarrow \infty}\frac{\sqrt[4]{n^2+1}-\sqrt{n+1}}{\sqrt[8]{n^2+1}+\sqrt[4]{n+1}}")
![=\displaystyle \lim_{n\rightarrow \infty}\frac{\sqrt{n^2+1}-(n+1)}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})}](http://s0.wp.com/latex.php?latex=%3D%5Cdisplaystyle+%5Clim_%7Bn%5Crightarrow+%5Cinfty%7D%5Cfrac%7B%5Csqrt%7Bn%5E2%2B1%7D-%28n%2B1%29%7D%7B%28%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%5B4%5D%7Bn%2B1%7D%29%28%5Csqrt%5B4%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%7Bn%2B1%7D%29%7D&bg=ffffff&fg=000000&s=0 "=\displaystyle \lim_{n\rightarrow \infty}\frac{\sqrt{n^2+1}-(n+1)}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})}")
![=\displaystyle \lim_{n\rightarrow \infty}\frac{(n^2+1)-(n+1)^2}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})(\sqrt{n^2+1}+(n+1))}](http://s0.wp.com/latex.php?latex=%3D%5Cdisplaystyle+%5Clim_%7Bn%5Crightarrow+%5Cinfty%7D%5Cfrac%7B%28n%5E2%2B1%29-%28n%2B1%29%5E2%7D%7B%28%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%5B4%5D%7Bn%2B1%7D%29%28%5Csqrt%5B4%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%7Bn%2B1%7D%29%28%5Csqrt%7Bn%5E2%2B1%7D%2B%28n%2B1%29%29%7D&bg=ffffff&fg=000000&s=0 "=\displaystyle \lim_{n\rightarrow \infty}\frac{(n^2+1)-(n+1)^2}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})(\sqrt{n^2+1}+(n+1))}")
![=\displaystyle \lim_{n\rightarrow \infty}\frac{-2n}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})(\sqrt{n^2+1}+(n+1))}](http://s0.wp.com/latex.php?latex=%3D%5Cdisplaystyle+%5Clim_%7Bn%5Crightarrow+%5Cinfty%7D%5Cfrac%7B-2n%7D%7B%28%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%5B4%5D%7Bn%2B1%7D%29%28%5Csqrt%5B4%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%7Bn%2B1%7D%29%28%5Csqrt%7Bn%5E2%2B1%7D%2B%28n%2B1%29%29%7D&bg=ffffff&fg=000000&s=0 "=\displaystyle \lim_{n\rightarrow \infty}\frac{-2n}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})(\sqrt{n^2+1}+(n+1))}")
![=\displaystyle \lim_{n\rightarrow \infty}\frac{-2}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})(\sqrt{1+\frac{1}{n^2}}+(1+\frac{1}{n}))}](http://s0.wp.com/latex.php?latex=%3D%5Cdisplaystyle+%5Clim_%7Bn%5Crightarrow+%5Cinfty%7D%5Cfrac%7B-2%7D%7B%28%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%5B4%5D%7Bn%2B1%7D%29%28%5Csqrt%5B4%5D%7Bn%5E2%2B1%7D%2B%5Csqrt%7Bn%2B1%7D%29%28%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bn%5E2%7D%7D%2B%281%2B%5Cfrac%7B1%7D%7Bn%7D%29%29%7D&bg=ffffff&fg=000000&s=0 "=\displaystyle \lim_{n\rightarrow \infty}\frac{-2}{(\sqrt[8]{n^2+1}+\sqrt[4]{n+1})(\sqrt[4]{n^2+1}+\sqrt{n+1})(\sqrt{1+\frac{1}{n^2}}+(1+\frac{1}{n}))}")
Or, you may think about that
![\sqrt[8]{n^2+1}-\sqrt[4]{n+1}](http://s0.wp.com/latex.php?latex=%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D-%5Csqrt%5B4%5D%7Bn%2B1%7D&bg=ffffff&fg=000000&s=0 "\sqrt[8]{n^2+1}-\sqrt[4]{n+1}")
![=\sqrt[8]{n^2+1}-\sqrt[8]{n^2}+\sqrt[8]{n^2}-\sqrt[4]{n+1}](http://s0.wp.com/latex.php?latex=%3D%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D-%5Csqrt%5B8%5D%7Bn%5E2%7D%2B%5Csqrt%5B8%5D%7Bn%5E2%7D-%5Csqrt%5B4%5D%7Bn%2B1%7D&bg=ffffff&fg=000000&s=0 "=\sqrt[8]{n^2+1}-\sqrt[8]{n^2}+\sqrt[8]{n^2}-\sqrt[4]{n+1}")
![=(\sqrt[8]{n^2+1}-\sqrt[8]{n^2})+(\sqrt[4]{n}-\sqrt[4]{n+1})](http://s0.wp.com/latex.php?latex=%3D%28%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D-%5Csqrt%5B8%5D%7Bn%5E2%7D%29%2B%28%5Csqrt%5B4%5D%7Bn%7D-%5Csqrt%5B4%5D%7Bn%2B1%7D%29&bg=ffffff&fg=000000&s=0 "=(\sqrt[8]{n^2+1}-\sqrt[8]{n^2})+(\sqrt[4]{n}-\sqrt[4]{n+1})")
and consider that both
![\sqrt[8]{n^2+1}-\sqrt[8]{n^2}](http://s0.wp.com/latex.php?latex=%5Csqrt%5B8%5D%7Bn%5E2%2B1%7D-%5Csqrt%5B8%5D%7Bn%5E2%7D&bg=ffffff&fg=000000&s=0 "\sqrt[8]{n^2+1}-\sqrt[8]{n^2}")
![\sqrt[4]{n}-\sqrt[4]{n+1}](http://s0.wp.com/latex.php?latex=%5Csqrt%5B4%5D%7Bn%7D-%5Csqrt%5B4%5D%7Bn%2B1%7D&bg=ffffff&fg=000000&s=0 "\sqrt[4]{n}-\sqrt[4]{n+1}")
tend to zero as 
[OT]
那早晨聽著手機傳來的新聞報導：「……新高中檢討後，將取消數學、體育、企業會計及財務管理三科……..」
@_@
「……的校本評核。」
但有家長高興地告訴子弟：「不用擔心了，數學將不會是主科！」這當然不涉及主流媒體誤導，純粹是家長不清楚何謂校本評核而已。
身為數學授課員，「看」到現今學生對數學的表現與態度，包括在顏冊或別的地方對數學和／或數學授課員的咒罵，希望取消數學科的學生（或家長）似乎佔大多數。
聞說「生活化」有助提升學習數學之興趣，今屆 DSE 數學考卷問「保安禁區」面積夠不夠大，也問公屋總樓面面積是否「供過於求」，使我有所聯想，不知我是否太政治敏感？
