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Mo Mai 29 17:29:54 CEST 2006

=head2 82. Hausaufgabe

=head3 Analysis-Buch Seite 115, Aufgabe 62

Berechne:

=over

=item a)

M<\lim\limits_{x \to \infty} \frac{e^{\sqrt{x}}}{x} = \infty;>

=item b)

M<\lim\limits_{x \to \infty} \frac{\sqrt{e^x}}{x} = \infty;>

=item c)

M<\lim\limits_{x \to -\infty} \frac{\sqrt{e^x}}{x} = 0;>

=item d)

M<\lim\limits_{x \to -\infty} x e^x = 0;>

=item e)

M<\lim\limits_{x \to \infty} \frac{\sqrt{e^x - 1190}}{e^x} = \lim\limits_{x \to
\infty} e^{-\frac{1}{2}x} = 0;>

=item f)

M<\lim\limits_{x \to \infty} \frac{e^x + e^{-x}}{e^x - e^{-x}} = \lim\limits_{x
\to \infty} \frac{e^x \left(1 + e^{-2x}\right)}{e^x \left(1 - e^{-2x}\right)} =
1;>

=item g)

M<\lim\limits_{x \to \infty} \frac{e^x + e^{-x}}{e^x - e^{-x}} = \lim\limits_{x
\to \infty} \frac{e^{-x} \left(e^{2x} + 1\right)}{e^{-x} \left(e^{2x} -
1\right)} = -1;>

=item h)

M<\lim\limits_{x \to -\infty} \frac{1}{\left(e^x - 1\right) \left(e^{x-2} -
1\right)} = \frac{1}{\left(-1\right) \left(-1\right)} = 1;>

=back