limit as x approaches infinity of (5x^2)/(e^{x^3)}

Lösung

x \to \infty lim (\frac{5 x ^{2}}{e ^{x^{3}}})

0

Schritte zur Lösung

x \to \infty lim (\frac{5 x ^{2}}{e ^{x^{3}}})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 5 \cdot x \to \infty lim (\frac{x ^{2}}{e ^{x^{3}}})

Wende Exponentenregel an:

\frac{1}{a ^{b}} = a^{- b}

\frac{x ^{2}}{e ^{x^{3}}} = (x^{2}) e^{- x^{3}}

= 5 \cdot x \to \infty lim (x^{2} e^{- x^{3}})

Umformung für L'Hopital

= 5 \cdot x \to \infty lim (\frac{x ^{2}}{\frac{1}{e ^{- x^{3}}}})

Wende das L'Hopital Theorem an

= 5 \cdot x \to \infty lim (\frac{2 x}{e ^{x^{3}} \cdot 3 x ^{2}})

Wende das L'Hopital Theorem an

= 5 \cdot x \to \infty lim (\frac{2}{3 ( 3 e ^{x^{3}} x ^{4} + 2 e ^{x^{3}} x )})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 5 \cdot \frac{2}{3} \cdot x \to \infty lim (\frac{1}{3 e ^{x^{3}} x ^{4} + 2 e ^{x^{3}} x})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

mit Einschränkung des unbestimmten Ausdrucks

= 5 \cdot \frac{2}{3} \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 3 e ^{x^{3}} x ^{4} + 2 e ^{x^{3}} x )}

x \to \infty lim (1) = 1

x \to \infty lim (3 e^{x^{3}} x^{4} + 2 e^{x^{3}} x) = \infty

= 5 \cdot \frac{2}{3} \cdot \frac{1}{\infty}

Vereinfache

5 \cdot \frac{2}{3} \cdot \frac{1}{\infty} : 0

= 0

x \to 0 - lim (\frac{x}{1 - cos ( x )})

x \to 0 lim (\frac{2 e ^{3 x} - 2}{ln ( x + 1 ) - 3 x ^{3}})

x \to 2 lim (x^{2} - 2 x + 5)

x \to 1 lim (- \frac{4}{x - 1})

x \to \infty lim ((x^{2} + 1)^{\frac{1}{x ^{2}}})