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

The limit as x approaches 0 of ((e^x-e^{-x}-2x))/((x-sin(x))) is 2

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limit as x approaches 0 of ((e^x-e^{-x}-2x))/((x-sin(x)))

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

2

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{e ^{0} + e ^{- 0}}{cos ( 0 )}

Simplify

\frac{e ^{0} + e ^{- 0}}{cos ( 0 )} : 2

= 2

y = \frac{1}{arctan ( x )}

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

\frac{d}{d x} (2 + x^{\frac{3}{2}})

\int_{- 2}^{2} 4 d x

\frac{\partial}{\partial y} (- x^{4} + 4 x y - y^{4} + 1)

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