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

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

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

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

0

Solution steps

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

\frac{- 2 sin ( 2 \cdot 0 ) + sin ( 0 )}{2} : 0

= 0

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