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 \(n\to\infty\)