What is the limit as x approaches 0 of (2e^{-x}-x^2+2x-2)/(4x^3) ?

The limit as x approaches 0 of (2e^{-x}-x^2+2x-2)/(4x^3) is -1/12

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limit as x approaches 0 of (2e^{-x}-x^2+2x-2)/(4x^3)

x \to 0 lim (\frac{2 e ^{- x} - x ^{2} + 2 x - 2}{4 x ^{3}})

- \frac{1}{12}

Decimal Notation

- 0.08333 \dots

Solution steps

x \to 0 lim (\frac{2 e ^{- x} - x ^{2} + 2 x - 2}{4 x ^{3}})

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

= \frac{1}{4} \cdot x \to 0 lim (\frac{2 e ^{- x} - x ^{2} + 2 x - 2}{x ^{3}})

Apply L'Hopital's Rule

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

= \frac{1}{4} \cdot x \to 0 lim (\frac{- 2 x - 2 e ^{- x} + 2}{3 x ^{2}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 e ^{- x} - 2}{6 x})

= \frac{1}{4} \cdot x \to 0 lim (\frac{2 e ^{- x} - 2}{6 x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 2 e ^{- x}}{6})

= \frac{1}{4} \cdot x \to 0 lim (\frac{- 2 e ^{- x}}{6})

Plug in the value

x = 0

= \frac{1}{4} \cdot \frac{- 2 e ^{- 0}}{6}

Simplify

\frac{1}{4} \cdot \frac{- 2 e ^{- 0}}{6} : - \frac{1}{12}

= - \frac{1}{12}

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What is the limit as x approaches 0 of (2e^{-x}-x^2+2x-2)/(4x^3) ?
The limit as x approaches 0 of (2e^{-x}-x^2+2x-2)/(4x^3) is -1/12