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limit as x approaches infinity of x*(arctan(e^x)-pi/2)

Lösung

x \to \infty lim (x \cdot (arctan (e^{x}) - \frac{π}{2}))

0

Schritte zur Lösung

x \to \infty lim (x (arctan (e^{x}) - \frac{π}{2}))

Umformung für L'Hopital

= x \to \infty lim (\frac{arctan ( e ^{x} ) - \frac{π}{2}}{\frac{1}{x}})

Wende das L'Hopital Theorem an

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

Vereinfache

\frac{\frac{e ^{x}}{e ^{2 x} + 1}}{- \frac{1}{x ^{2}}} : - \frac{e ^{x} x ^{2}}{e ^{2 x} + 1}

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

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{- e ^{x} x ^{2} - 2 e ^{x} x}{2 e ^{2 x}})

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{- e ^{x} x ^{2} - 4 e ^{x} x - 2 e ^{x}}{4 e ^{2 x}})

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{- e ^{x} x ^{2} - 6 e ^{x} x - 6 e ^{x}}{8 e ^{2 x}})

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{- e ^{x} x ^{2} - 8 e ^{x} x - 12 e ^{x}}{16 e ^{2 x}})

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{- e ^{x} x ^{2} - 10 e ^{x} x - 20 e ^{x}}{32 e ^{2 x}})

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{- e ^{x} x ^{2} - 12 e ^{x} x - 30 e ^{x}}{64 e ^{2 x}})

Wende das L'Hopital Theorem an

= x \to \infty lim (\frac{- e ^{x} x ^{2} - 14 e ^{x} x - 42 e ^{x}}{128 e ^{2 x}})

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

= \frac{1}{128} \cdot x \to \infty lim (\frac{- e ^{x} x ^{2} - 14 e ^{x} x - 42 e ^{x}}{e ^{2 x}})

Dividiere durch

e^{2 x} : - \frac{x ^{2}}{e ^{x}} - \frac{14 x}{e ^{x}} - \frac{42}{e ^{x}}

= \frac{1}{128} \cdot x \to \infty lim (- \frac{x ^{2}}{e ^{x}} - \frac{14 x}{e ^{x}} - \frac{42}{e ^{x}})

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

mit Einschränkung des unbestimmten Ausdrucks

= \frac{1}{128} (- x \to \infty lim (\frac{x ^{2}}{e ^{x}}) - x \to \infty lim (\frac{14 x}{e ^{x}}) - x \to \infty lim (\frac{42}{e ^{x}}))

x \to \infty lim (\frac{x ^{2}}{e ^{x}}) = 0

x \to \infty lim (\frac{14 x}{e ^{x}}) = 0

x \to \infty lim (\frac{42}{e ^{x}}) = 0

= \frac{1}{128} (- 0 - 0 - 0)

Vereinfache

= 0

\frac{d ^{2}}{d x ^{2}} (e^{- 2 x})

\int \frac{1}{( x ^{4} - 1 ) ^{2}} d x

p a r i t y y = x + x + x

\int \frac{1}{v ^{2} + 2 v} d v

\frac{d}{d x} (- (3 x + 2) cos (2 x))