Problem 292 (difficulty: 4/10)

Calculate the limit of the following sequences.

 (1)  \(\displaystyle \sqrt[n]{2n+\sqrt n}\),      (2)  \(\displaystyle \frac{n^7-6n^6+5n^5-n-1}{n^3+n^2+n+1}\),      (3)  \(\displaystyle \frac{n^3+n^2\sqrt n -\sqrt n +1}{2n^3-6n+\sqrt n -2}\),      (4)  \(\displaystyle \sqrt[n]{\frac{1}{n}-\frac{2}{n^2}}\),
 (5)  \(\displaystyle \sqrt[n]{2^n+3^n}\),      (6)  \(\displaystyle \frac{\sqrt{2n+1} }{\sqrt {3n} +4}\),      (7)  \(\displaystyle \log \frac{n+1}{n+2}\),      (8)  \(\displaystyle \frac{7^n-7^{-n}}{7^n+7^{-n}}\),
 (9)  \(\displaystyle \frac{(2n+3)^5\cdot (18n+17)^{15}}{(6n+5)^{20}}\),      (10)  \(\displaystyle \frac{\sqrt{4n^2+2n+100}}{\sqrt[3]{6n^3-7n^2+2}}\),     (11)  \(\displaystyle \frac{\sqrt[4]{n^3+6}}{\sqrt[3]{n^2+3n-2}}\),
 (12)  \(\displaystyle n\cdot(\sqrt {n+1} -\sqrt n )\),      (13)  \(\displaystyle \frac{2^n+5^n}{3^n+1}\),      (14)  \(\displaystyle n\cdot(\sqrt {n^2+n} -\sqrt {n^2-n})\).

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