Задачи для самостоятельного решения

1. $$\lim_{n \to \infty}{\frac{(n+1)(n+2)(n+3)}{3n^3+3n+7}}$$

2. $$\lim_{n \to \infty}{\left( \frac{n^2+1}{2n+1}-\frac{3n^2+1}{6n+1}\right)}$$

3. $$\lim_{n \to \infty}{\frac{3^n+4^n+5^n}{5^{n+1}+6^{n+1}}}$$

4. $$\lim_{n \to \infty}{\frac{1+2+\ldots+n}{n^2-1}}$$

5. $$\lim_{n \to \infty}{\frac{1^2+2^2+\ldots+n^2}{n^3}}$$

6. $$\lim_{n \to \infty}{\left( \frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\cdots+\frac{1}{n(n+1)} \right)}$$

7. $$\lim_{n \to \infty}{\frac{\sqrt{n^2+n}-\sqrt{9n^2+2n}}{\sqrt[3]{n^3+1}-\sqrt[3]{8n^3+2}}}$$

8. $$\lim_{n \to \infty}{\frac{3n+2\cos^2 n!}{n+5}}$$

9. $$\lim_{n \to \infty}{\left( \frac{n^2+1}{n^2-1} \right)^{n^2}}$$

10. $$\lim_{n \to \infty}{\frac{n!+(n+2)!}{n\cdot ((n+1)!+n!)}}$$

11. $$\lim_{n \to \infty}{\sqrt{2}\cdot \sqrt[4]{2}\cdot \sqrt[8]{2}\cdot \ldots \cdot \sqrt[2^n]{2}}$$

12. $$\lim_{n \to \infty}{\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\ldots+\frac{1}{\sqrt{n^2+n}} \right)}$$