What is the limit as x approaches 0+of 1/(e^x-1)-1/x ?

The limit as x approaches 0+of 1/(e^x-1)-1/x is -1/2

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limit as x approaches 0+of 1/(e^x-1)-1/x

x \to 0 + lim (\frac{1}{e ^{x} - 1} - \frac{1}{x})

- \frac{1}{2}

Decimal Notation

- 0.5

Solution steps

x \to 0 + lim (\frac{1}{e ^{x} - 1} - \frac{1}{x})

Simplify

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

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

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{- e ^{x} + 1}{e ^{x} - 1 + e ^{x} x})

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{- e ^{0}}{e ^{0} \cdot 0 + 2 e ^{0}}

Simplify

\frac{- e ^{0}}{e ^{0} \cdot 0 + 2 e ^{0}} : - \frac{1}{2}

= - \frac{1}{2}

y = 4 - x^{2}, y = 0

cos (3 t) - cos (7 t)

x^{3} + 3

\int (x^{3} - \frac{1}{x ^{3}})^{2} d x

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

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