Serlo: EN: Examples for limits

{{#invoke:Mathe für Nicht-Freaks/Seite|oben}}

Some important limits

The following limits are fundamental building bricks, which you can use to find the limits for a lot of other sequences. You should know them by heart - and have a figure in mind what they look like and how they converge:

• $\underset{n\to \mathrm{\infty }}{\lim }c=c$ for all $c\in ℝ$
• $\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}=0$
• $\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n^k}=0$ for all $k\in \mathbb{N}$
• $\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{\sqrt[k]{n}}=0$ for all $k\in \mathbb{N}$
• $\underset{n\to \mathrm{\infty }}{\lim }{q}^n=0$ for all $q\in ℝ$ with $|q|<1$
• $\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{c}=1$ for all $c>0$
• $\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{n}=1$
• $\underset{n\to \mathrm{\infty }}{\lim }\frac{n^k}{z^n}=0$ for all $k\in \mathbb{N}$ and $z\in ℝ$ with $|z|>1$
• $\underset{n\to \mathrm{\infty }}{\lim }{n}^k{q}^n=0$ for all $k\in \mathbb{N}$ and $q\in ℝ$ with $|q|<1$
• $\underset{n\to \mathrm{\infty }}{\lim }\frac{z^n}{n!}=0$ for all $z\in ℝ$ with $|z|>1$
• $\underset{n\to \mathrm{\infty }}{\lim }\frac{n!}{n^n}=0$
• $\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n=e$

Or intuitively, polynomial beats constant, exponential beats polynomial, factorial beats exponential and $n^n$ beats them all.

Ratio - power series / geometric series Vorlage:Anker

In simple words: "exponential beats polynomial":

Ratio - geometric series / factorial sequences

In simple words: "factorial beats exponential":

Ratio - factorial / $n^n$

In simple words: "$n^n$ beats factorial":

{{#invoke:Mathe für Nicht-Freaks/Seite|unten}}