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

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

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

0

Solution steps

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

\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})

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

= 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}})

Divide by

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)

With the exception of indeterminate form

= \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)

Simplify

= 0

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\frac{\partial}{\partial y} (- e^{- x} sin (yz))

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What is the limit as x approaches infinity of x*(arctan(e^x)-pi/2) ?
The limit as x approaches infinity of x*(arctan(e^x)-pi/2) is 0