Tabulky 7.2 a 7.3 shrnují „známé“ limitní vztahy, které jsme odvodili v této a předchozí kapitole. Tyto výsledky lze v příkladech v písemkách považovat za známé (pokud ovšem zadání/otázka nemíří přímo na jejich odvození).
Důležité limity | Hodnota | Parametry |
---|---|---|
\(\displaystyle\lim_{n\to\infty} c\) | \(c\) | \(c \in \R\) |
\(\displaystyle\lim_{n\to\infty} n^a\) | \(\begin{cases} +\infty & a > 0, \\ 1 & a = 0, \\ 0, & a < 0. \end{cases}\) | \(a \in \R\) |
\(\displaystyle\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k}\) | \(+\infty\) | |
\(\displaystyle\lim_{n\to\infty} \sqrt[n]{n}\) | \(1\) | |
\(\displaystyle\lim_{n\to\infty} \sqrt[n]{c}\) | \(1\) | \(c \in (0, +\infty)\) |
\(\displaystyle\lim_{n\to\infty} \sqrt[n]{n!}\) | \(+\infty\) | |
\(\displaystyle\lim_{n\to\infty} a^n\) | \(\begin{cases} 0, & |a| < 1, \\ 1, & a = 1, \\ +\infty, & a > 1, \\ \text{neexistuje}, & a \leq -1. \end{cases}\) | \(a\in\R\) |
\(\displaystyle\lim_{n\to\infty} \left(1 + \frac{1}{n} \right)^n\) | \(\mathrm{e}\) |
Důležité limity | Hodnota | Parametry |
---|---|---|
\(\displaystyle\lim_{x\to a} c\) | \(c\) | \(c \in \R\), \(a \in \overline{\R}\) |
\(\displaystyle\lim_{x\to a} x\) | \(a\) | \(a \in \overline{\R}\) |
\(\displaystyle\lim_{x\to a\pm} \frac{1}{(x-a)^k}\) | \(\begin{cases} \pm\infty, & k \ \text{liché}, \\ +\infty, & k \ \text{sudé}.\end{cases}\) | \(a \in \R\), \(k \in \N\) |
\(\displaystyle\lim_{x\to a} |x|\) | \(|a|\) | \(a \in \overline{\R}\), kde \(\left|+\infty\right| = \left|-\infty\right| = +\infty\) |
\(\displaystyle\lim_{x\to0\pm} \sgn(x)\) | \(\pm 1\) | |
\(\displaystyle\lim_{x\to a} \sqrt[k]{x}\) | \(\sqrt[k]{a}\) | liché \(k \in \N\), \(a \in \overline{\R}\) |
\(\displaystyle\lim_{x\to a} \sqrt[k]{x}\) | \(\sqrt[k]{a}\) | sudé \(k \in \N\), \(a \in \langle 0, +\infty) \cup \{+\infty\}\) |
\(\displaystyle\lim_{x\to a} P(x)\) | \(P(a)\) | \(a \in \R\), \(P\) polynom |
\(\displaystyle\lim_{x\to0} \frac{e^x - 1}{x}\) | \(1\) | |
\(\displaystyle\lim_{x\to0} \frac{\ln(1+x)}{x}\) | \(1\) | |
\(\displaystyle\lim_{x\to0} \frac{\sin(x)}{x}\) | \(1\) | |
\(\displaystyle\lim_{x\to a} \sin(x)\) | \(\sin(a)\) | \(a \in \R\) |
\(\displaystyle\lim_{x\to a} \cos(x)\) | \(\cos(a)\) | \(a \in \R\) |
\(\displaystyle\lim_{x\to a} \mathrm{e}^x\) | \(\mathrm{e}^a\) | \(a \in \R\) |
\(\displaystyle\lim_{x\to +\infty} \mathrm{e}^x\) | \(+\infty\) | |
\(\displaystyle\lim_{x\to -\infty} \mathrm{e}^x\) | \(0\) | |
\(\displaystyle\lim_{x\to a} \ln(x)\) | \(\ln(a)\) | \(a \in (0, +\infty)\) |
\(\displaystyle\lim_{x\to +\infty} \ln(x)\) | \(+\infty\) | |
\(\displaystyle\lim_{x\to0+} \ln(x)\) | \(-\infty\) | |
\(\displaystyle\lim_{x\to\pm\infty} \left(1 + \frac{1}{x}\right)^x\) | \(\mathrm{e}\) | |
\(\displaystyle\lim_{x\to\pm\infty} \left(1+\frac{\alpha}{x}\right)^x\) | \(\mathrm{e}^\alpha\) | \(\alpha \in \R\) |