Problem 291 (difficulty: 4/10)

Calculate the limit of the following sequences.

 (1)  \(\displaystyle \frac{3n+16}{4n-25}\),      (2)  \(\displaystyle n\cdot\left( \sqrt {1+\frac{1}{n}}-1 \right)\),      (3)  \(\displaystyle \frac{1}{n}\cdot \frac{n^2+1}{n^3+1}\),      (4)  \(\displaystyle \frac{5-2n^2}{4+n}\),
 (5)  \(\displaystyle \frac{\sin (n)+n}{n}\),      (6)  \(\displaystyle \frac{2n^3+3\sqrt n }{1-n^3}\),      (7)  \(\displaystyle \sqrt[n]{n+5^n}\),      (8)  \(\displaystyle \frac{2^n+n!}{n^n-n^{1000}}\),
 (9)  \(\displaystyle \sqrt[n]{n^n-5^n}\),      (10)  \(\displaystyle \frac{\sin(n)}{n} \),      (11)  \(\displaystyle \frac{5n^2+(-1)^n}{8n}\),      (12)  \(\displaystyle \frac{6n+2n^2\cdot (-1)^n}{n^2}\).

Give me another random problem!