What is the limit as x approaches 0 of (sinh(x)-x)/(x^3) ?

The limit as x approaches 0 of (sinh(x)-x)/(x^3) is 1/6

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limit as x approaches 0 of (sinh(x)-x)/(x^3)

x \to 0 lim (\frac{sinh ( x ) - x}{x ^{3}})

\frac{1}{6}

Decimal

0.16666 \dots

Solution steps

x \to 0 lim (\frac{sinh ( x ) - x}{x ^{3}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{cosh ( x ) - 1}{3 x ^{2}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{sinh ( x )}{6 x})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{cosh ( x )}{6})

Plug in the value

x = 0

= \frac{cosh ( 0 )}{6}

cosh (0) = 1

= \frac{1}{6}

What is the limit as x approaches 0 of (sinh(x)-x)/(x^3) ?
The limit as x approaches 0 of (sinh(x)-x)/(x^3) is 1/6