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

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

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

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

\infty

Solution steps

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

Simplify

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

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

Simplify

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

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

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

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

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

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

= 1 \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

\frac{d}{d x} (13 arctan (x))

(3 x^{4} + 4)^{2}

y^{^{''}} + 4 y^{^{'}} + 3 y = 6 e^{- x}

\int \frac{1}{a x + b} d x

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = \frac{e ^{x}}{x ^{2} + 1}

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