Comparison of functions O(f) and o(f).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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