notation
the limit (Grenzwert auch Limes 1) of an expression an as n approaches infinity is written as $${\lim_{n \to \infty} a_n}$$
n is the index.
example
investigating the sequence $${(a_n) = \frac{2n + 3}{n}}$$ as $${n \rightarrow \infty }$$
plotter using jsxgraph
link to a sequence plotter where you can enter different sequences and investigate the behaviour as the index n increases
entering higher values for n will show that the terms of the sequence are approaching the value 2.
$${\lim_{n \to \infty} \frac{2n + 3}{n} = 2}$$ Dies wird gelesen als … $${ \text{“Der Limes von} \frac{2n + 3}{n} \text{für n gegen unendlich ist gleich 2”}}$$
formal definition of a limit (Grenzwert)
A sequence of real numbers (an) converges to a real number L if, for all $${\epsilon > 0}$$, there exists a natural number N such that for all $${n \geq N}$$ we have $${|a_n - L| < \epsilon}$$.
converging - konvergent
a sequence that has a real limit is said to converge.
diverging - divergent
a sequence that does not converge to some finite limit is said to diverge.
uneigentlicher Grenzwert
If, for any large number K, there is an index number n after which all the following terms are larger than K then the terms increase beyond all limits.
We write $${\lim_{n \to \infty} a_n = \infty}$$
Theorem (Satz):
Suppose that the sequences (an) and (bn) converge to limits A and B, respectively, and c is a constant. Then $${\lim_{n \to \infty} c = c}$$ $${\lim_{n \to \infty} ca_n = cA}$$ $${\lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} (a_n) + \lim_{n \to \infty} (b_n) = A + B}$$ $${\lim_{n \to \infty} (a_n - b_n) = \lim_{n \to \infty} (a_n) - \lim_{n \to \infty} (b_n) = A - B}$$ $${\lim_{n \to \infty} (a_nb_n) = \lim_{n \to \infty} (a_n) \cdot \lim_{n \to \infty} (b_n) = AB}$$ $${\lim_{n \to \infty} (\frac{a_n}{b_n}) = \frac{\lim\limits_{n \to \infty } (a_n)}{\lim\limits_{n \to \infty} (b_n)} = \frac{A}{B} \space \space (\text{if} \space B \neq 0)}$$
examples
$${(a_n) = \frac{n}{2n+1}}$$
$${(a_n) = (-1)^{n+1}\frac{n}{2n+1}}$$
$${(a_n) = (-1)^{n+1}\frac{1}{n}}$$
$${(a_n) = 8 -2n}$$
Theorem (Satz):
A sequence converges to a limit L if and only if the sequences of even-numbered terms and odd-numbered terms both converge to L.
The Squeezing Theorem for Sequences - Einschachtelungssatz
Let (an), (bn) and (cn) be sequences such that $${a_n \leq b_n \leq c_n }$$ for all values of n beyond some index N. If the sequences (an) and (cn) have a common limit L as $${n \rightarrow \infty}$$ , then (bn) also has the limit L as $${n \rightarrow \infty}$$.
limit of a function - Der Grenzwert einer Funktion
Limits are used to investigate functions at the boundaries of their domain (Definitionsbereich). There are two types of limiting processes.
$${x \rightarrow \infty \text{ or }x \rightarrow - \infty}$$ and $${x \rightarrow x_0}$$
function plotter
A link to a relatively simple function plotter adapted from jsxgraph.org.
example
How do the function values of $${f(x) = \frac{3x + 1}{x}, x > 0}$$ develop as x gets larger and larger? Sketch the graph of the function an comment on it.
The graph approaches the line y = 3 from above as $${x \rightarrow \infty }$$ This line is called an asymptote.
- infinity and infinity
investigating a function at a specific point
approaching from below and above
linksseitiger und rechsseitiger Grenzwert
exercises
link to practice questions with answers
-
lateinisch für befestigete Grenzlinie, Grenze, Grenzwall, Querweg oder Grenzlinie. Der Grenzwall an der römischen Reichsgrenze wurde auch als Limes bezeichnet. ↩︎