Zum Inhalt springen
Version 7.1 von Tina Müller am 2024/10/15 09:51
Zeige letzte Bearbeiter
1 (% style="list-style: alphastyle" %)
2 1. (((Randverhalten: Verhalten im Unendlichen
3 1) Verhalten gegen plus Unendlich ({{formula}}+\infty{{/formula}})
4 (% class="border" %)
5 |={{formula}}x{{/formula}}| {{formula}}+1{{/formula}}| {{formula}}+10{{/formula}}| {{formula}}+100{{/formula}}| {{formula}}+1000{{/formula}}| {{formula}}+10^6{{/formula}}| {{formula}}+10^9{{/formula}}| {{formula}}+10^{12}{{/formula}}|{{formula}}+10^{+\infty}{{/formula}}
6 |={{formula}}f(x){{/formula}}|1|{{formula}}\frac{1}{10}{{/formula}}|{{formula}}\frac{1}{100}{{/formula}}|{{formula}}\frac{1}{1000}{{/formula}}|{{formula}}\frac{1}{1000000}{{/formula}}|{{formula}}\frac{1}{1000000000}{{/formula}}|{{formula}}\frac{1}{1000000000000}{{/formula}}|0
7
8 2) Verhalten gegen minus Unendlich ({{formula}}-\infty{{/formula}})
9 (% class="border" %)
10 |={{formula}}x{{/formula}}| {{formula}}-1{{/formula}}| {{formula}}-10{{/formula}}| {{formula}}-100{{/formula}}| {{formula}}-1000{{/formula}}| {{formula}}-10^6{{/formula}}| {{formula}}-10^9{{/formula}}|{{formula}}-10^{12}{{/formula}}|{{formula}}-10^{-\infty}{{/formula}}
11 |={{formula}}f(x){{/formula}}|{{formula}}-1{{/formula}}|{{formula}}-\frac{1}{10}{{/formula}}|{{formula}}-\frac{1}{100}{{/formula}}|{{formula}}-\frac{1}{1000}{{/formula}}|{{formula}}-\frac{1}{1000000}{{/formula}}|{{formula}}\frac{1}{1000000000}{{/formula}}|{{formula}}\frac{1}{1000000000000}{{/formula}}|0
12 )))
13 1. (((Randverhalten: Verhalten nahe der Definitionslücke ({{formula}}x \approx 0{{/formula}})
14 1) Verhalten links bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x<0{{/formula}})
15 (% class="border" %)
16 |={{formula}}x{{/formula}}| {{formula}}-1{{/formula}}| {{formula}}-0,1{{/formula}}| {{formula}}-0,01{{/formula}}| {{formula}}-0,001{{/formula}}| {{formula}}-10^{-6}{{/formula}}| {{formula}}-10^{-9}{{/formula}}| {{formula}}-10^{-12}{{/formula}}|0
17 |={{formula}}f(x){{/formula}}|{{formula}}-1{{/formula}}|{{formula}}-10{{/formula}}|{{formula}}-100{{/formula}}|{{formula}}-1000{{/formula}}|{{formula}}-10^6{{/formula}}| {{formula}}-10^9{{/formula}}|{{formula}}-10^{-12}{{/formula}}|{{formula}}-\infty{{/formula}}
18
19 2) Verhalten rechts bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x>0{{/formula}})
20 (% class="border" %)
21 |={{formula}}x{{/formula}}| {{formula}}+1{{/formula}}| {{formula}}+0,1{{/formula}}| {{formula}}+0,01{{/formula}}| {{formula}}+0,001{{/formula}}| {{formula}}+10^{-6}{{/formula}}| {{formula}}+10^{-9}{{/formula}}| {{formula}}+10^{-12}{{/formula}}|0
22 |={{formula}}f(x){{/formula}}|{{formula}}1{{/formula}}|{{formula}}10{{/formula}}|{{formula}}100{{/formula}}|{{formula}}1000{{/formula}}|{{formula}}10^6{{/formula}}| {{formula}}10^9{{/formula}}|{{formula}}10^{-12}{{/formula}}|{{formula}}\infty{{/formula}}
23 )))
24
25 c.) Zur Symmetrie:
26 Die Funktion {{formula}} f(x) = \frac{1}{x} {{/formula}} ist punktsymmetrisch zur Ursprungsgeraden, da gilt:
27 {{formula}}
28 f(-x) = -f(x)
29 {{/formula}}
30 Das bedeutet, dass die Funktion eine Symmetrie bezüglich des Ursprungs hat.
31
32 d.) Zum Randverhalten:
33 {Verhalten im Unendlichen}
34 {{formula}} \( \lim_{x \to +\infty} f(x) = 0 \){{/formula}}
35 {{formula}} \( \lim_{x \to -\infty} f(x) = 0 \){{/formula}}
36
37 {Verhalten nahe der Definitionslücke}
38 {{formula}} \( \lim_{x \to 0^-} f(x) = -\infty \){{/formula}}
39 {{formula}}\( \lim_{x \to 0^+} f(x) = +\infty \){{/formula}}
40
41 1. Bestimme {{formula}}g(y){{/formula}} für {{formula}}y=g(x){{/formula}} und {{formula}}x\in \mathbb{R}^*{{/formula}}.