1. Nullstellen berechnen

Die Nullstellen von $ff$ sind die Nullstellen des Zählers, wenn diese ungleich $-2-2$ sind ($-2-2$ ist nicht in ${\mathbb{D}}_{\mathbb{f}}\backslash mathbb\{D\_f\}$ enthalten).

$\Rightarrow x^2-9=(x+3)\cdot (x-3)=0\backslash ;\backslash Rightarrow\backslash ;x^2-9=(x+3)\backslash cdot(x-3)=0$

Verwende den Satz vom Nullprodukt:

$\Rightarrow x+3=0\Rightarrow x_1=-3\backslash Rightarrow\backslash ;\; x+3=0\; \backslash ;\backslash Rightarrow\backslash ;x\_1=-3$ und $x-3=0\Rightarrow x_2=+3x-3=0\backslash ;\backslash Rightarrow\backslash ;x\_2=+3$

Die Funktion $ff$ hat die beiden Nullstellen $x_1=-3x\_1=-3$ und $x_2=+3x\_2=+3$.

Koordinaten des Schnittpunkts des Graphen von $ff$ mit der y-Achse

Berechne $f(0)f(0)$:

$\Rightarrow f(0)={\displaystyle \frac{0^2-9}{0+2}}={\displaystyle \frac{-9}{2}}=-\mathrm{4,5}\backslash ;\backslash Rightarrow\backslash ;f(0)=\backslash dfrac\{0^2-9\}\{0+2\}=\backslash dfrac\{-9\}\{2\}=-4\{,\}5$

Es ist${\displaystyle \underset{x\to -\mathrm{\infty }}{\lim }\frac{\stackrel{\to +\mathrm{\infty }}{\overbrace{x^2-9}}}{\underset{\to -\mathrm{\infty }}{\underbrace{x+2}}}=-\mathrm{\infty }}\backslash displaystyle\backslash lim\_\{x\backslash rightarrow\; -\backslash infty\}\backslash dfrac\{\backslash overbrace\{x^2-9\}^\{\backslash rightarrow\; +\backslash infty\}\}\{\backslash underbrace\{x+2\}\_\{\backslash rightarrow\; -\backslash infty\}\}=-\backslash infty$ und ${\displaystyle \underset{x\to +\mathrm{\infty }}{\lim }\frac{\stackrel{\to +\mathrm{\infty }}{\overbrace{x^2-9}}}{\underset{\to +\mathrm{\infty }}{\underbrace{x+2}}}=+\mathrm{\infty }}\backslash displaystyle\backslash lim\_\{x\backslash rightarrow\; +\backslash infty\}\backslash dfrac\{\backslash overbrace\{x^2-9\}^\{\backslash rightarrow\; +\backslash infty\}\}\{\backslash underbrace\{x+2\}\_\{\backslash rightarrow\; +\backslash infty\}\}=+\backslash infty$