Справочник по математике

Пределы

к содержанию справочника

1. $$\log_a{n}\prec n^{p}\prec a^{n}\prec n{!}$$

2. $$\lim_{n \to \infty}\left(1+\displaystyle\frac{1}{n}\right)^n=e$$

3. $$\lim_{n \to \infty}\sqrt[n]{n}=1$$

4. $$\lim_{n \to \infty}\displaystyle\frac{1}{\sqrt[n]{n!}}=0$$

5. $$\lim_{n \to \infty}\displaystyle\frac{n}{\sqrt[n]{n!}}=e$$

6. $$\lim_{x \to 0}\displaystyle\frac{\sin{x}}{x}=1$$

7. $$\lim_{x \to 0}\displaystyle\frac{1-\cos{x}}{x^2}=\frac{1}{2}$$

8. $$\lim_{x \to 0}\left(1+x\right)^{\frac{1}{x}}=e$$

9. $$\lim_{x \to \infty}\left(1+\displaystyle\frac{1}{x}\right)^x=e$$

10. $$\lim_{x \to 0}\displaystyle\frac{\ln(1+x)}{x}=1$$

11. $$\lim_{x \to 0}\displaystyle\frac{\log_{a}(1+x)}{x}=\frac{1}{\ln{a}}$$

12. $$\lim_{x \to 0}\displaystyle\frac{e^x-1}{x}=1$$

13. $$\lim_{x \to 0}\displaystyle\frac{a^x-1}{x}=\ln{a}$$

14. $$\lim_{x \to 0}\displaystyle\frac{(1+x)^{p}-1}{x}=p$$

15. $$\lim_{x \to +0}x^{p}\ln{x}=0 \quad (p>0)$$

16. $$\lim_{x \to +\infty}\displaystyle\frac{\ln^{q}{x}}{x^{p}}=0 \quad (p>0)$$

17. $$\lim_{x \to +0}x^{x}=1$$

18. $$\lim_{x \to 0}\displaystyle\frac{\mathrm{sh}{x}}{x}=1$$

19. $$\lim_{x \to 0}\displaystyle\frac{\mathrm{ch}{x}-1}{x^2}=\frac{1}{2}$$

20. $$\lim_{x \to 0}\displaystyle\frac{\mathrm{th}{x}}{x}=1$$

Сравнение функций

$f(x)=o(g(x))$ при $x\to a$, если $\displaystyle\lim_{x \to a}\displaystyle\frac{f(x)}{g(x)}=0$

Локально эквивалентные функции

$f(x)\sim g(x)$ при $x\to a$, если $\displaystyle\lim_{x \to a}\displaystyle\frac{f(x)}{g(x)}=1$

Эквивалентности при $x\to 0$

1. $\sin{x}\sim x$
2. $\mathrm{tg}{x}\sim x$
3. $1-\cos{x}\sim\displaystyle\frac{x^2}{2}$
4. $\mathrm{arcsin}{x}\sim x$
5. $\mathrm{arctg}{x}\sim x$
6. $e^x-1\sim x$
7. $a^x-1\sim x\ln{a}$
8. $\ln(1+x)\sim x$
9. $\ln(x+\sqrt{x^2+1})\sim x$
10. $(1+x)^p-1\sim px$
11. $\mathrm{sh}{x}\sim x$
12. $\mathrm{ch}{x}-1\sim\displaystyle\frac{x^2}{2}$

Формула Стирлинга

$n!\sim\sqrt{2\pi n}\left(\displaystyle\frac{n}{e}\right)^n$ при $n\to\infty$