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

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

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

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

1

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

= \frac{e ^{0} + e ^{- 0}}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

Simplify

\frac{e ^{0} + e ^{- 0}}{2 cos ( 0 ) - 0 \cdot sin ( 0 )} : 1

= 1

\int \frac{1}{x ^{2} + 8 x + 65} d x

\int e^{11 x} d x

\int (cos (x)) d x

x \to \frac{1}{4} lim (sin (\frac{π}{x}))

\frac{d}{d x} ((7 x + 1)^{2})

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