What is the limit as x approaches-infinity of 1-e^{-x}-xe^{-x} ?

The limit as x approaches-infinity of 1-e^{-x}-xe^{-x} is infinity

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limit as x approaches-infinity of 1-e^{-x}-xe^{-x}

x \to - \infty lim (1 - e^{- x} - x e^{- x})

\infty

Solution steps

x \to - \infty lim (1 - e^{- x} - x e^{- x})

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

1 - e^{- x} - x e^{- x} = x (\frac{1}{x} - \frac{e ^{- x}}{x} - e^{- x})

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

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

With the exception of indeterminate form

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

x \to - \infty lim (x) = - \infty

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

= (- \infty) (- \infty)

Apply Infinity Property:

(- \infty) (- \infty) = \infty

= \infty

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What is the limit as x approaches-infinity of 1-e^{-x}-xe^{-x} ?
The limit as x approaches-infinity of 1-e^{-x}-xe^{-x} is infinity