Question

Bestimmen Sie im Falle der Existenz folgende Grenzwerte:

a) \( \lim \limits_{x \rightarrow \infty} \frac{\ln x}{x} \)
b) \( \lim \limits_{x \downarrow 0}(x \cdot \ln x) \)
c) \( \lim \limits_{x \downarrow 0} x^{x} \)
d) \( \lim \limits_{x \rightarrow \infty} x^{1 / x} \)
e) \( \lim \limits_{n \rightarrow \infty} n(\sqrt[n]{x}-1) \quad( \) für \( x>0) \)

Answer 1

a) lim _x→∞ ln(x) / x = lim _x→∞ 1/x / 1 = 0

b) lim _x→0+ x * ln(x) =  lim _x→0+ ln(x) / (1/x) = lim _x→0+ 1/x / (-1/x^2) = lim _x→0+ -x = 0

c) lim _x→0+ x^x = lim _x→0+ e^{x * lnx} = e^{0} = 1

d) lim _x→∞ x^{1/x} = lim _x→∞ e^{1/x * ln x} = e^0 = 1

e) lim _n→∞ n(x^{1/n} - 1) = lim _z→0 1/z * (x^z - 1) = lim _z→0 (x^z - 1) / z = lim _z→0 x^z * ln(x) / 1 = ln(x)