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

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

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

x \to 0 lim (\frac{2}{x} - \frac{2}{e ^{x} - 1})

1

Solution steps

x \to 0 lim (\frac{2}{x} - \frac{2}{e ^{x} - 1})

Simplify

\frac{2}{x} - \frac{2}{e ^{x} - 1} : \frac{2 ( e ^{x} - 1 ) - 2 x}{x ( e ^{x} - 1 )}

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 e ^{x} - 2}{e ^{x} - 1 + e ^{x} x})

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= 1

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