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# Сравнение функций $O(f)$ и $o(f).$

1. $o(Cf)=o(f),\quad C-const.$

2. $o(f)+o(f)=o(f).$

3. $o(f)o(g)=o(fg).$

4. $o(o(f))=o(f).$

5. $O(O(f))=O(f).$

6. $o(O(f))=o(f).$

7. $O(o(f))=o(f).$

8. $o(f)O(f)=o(f).$

9. $o(f)+O(f)=O(f).$

$n\in N,\,\, k\in N,\,\, n>k.$

10. $o(x^n)+o(x^k)=o(x^k),\quad x\rightarrow 0.$

11. $o(x^n)+o(x^k)=o(x^n),\quad x\rightarrow\infty.$

12. $O(x^n)+O(x^k)=O(x^k),\quad x\rightarrow 0.$

13. $O(x^n)+O(x^k)=O(x^n),\quad x\rightarrow\infty.$

14. $O(x^n)O(x^k)=O(x^{n+k}),\quad x\rightarrow 0, \, x\rightarrow \infty.$

15. $f(x)\sim g(x), x\rightarrow x_0\Leftrightarrow f(x)=g(x)+o(g(x)),\quad x\rightarrow x_0.$

16. $\left\{\begin{array}{lcl}f(x)\sim f_1(x),\,\, x\rightarrow x_0,\\g(x)\sim g_1(x),\,\, x\rightarrow x_0,\\\exists\lim\limits_{x\rightarrow x_0}\frac{f_1(x)}{g_1(x)},\end{array}\right.\Rightarrow \lim\limits_{x\rightarrow x_0}\frac{f(x)}{g(x)}=\lim\limits_{x\rightarrow x_0}\frac{f_1(x)}{g_1(x)}.$