What is the limit as x approaches 0 of ((x-xcos(x)))/(sin^2(3x)) ?

The limit as x approaches 0 of ((x-xcos(x)))/(sin^2(3x)) is 0

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limit as x approaches 0 of ((x-xcos(x)))/(sin^2(3x))

x \to 0 lim (\frac{( x - x cos ( x ) )}{sin ^{2} ( 3 x )})

0

Solution steps

x \to 0 lim (\frac{x - x cos ( x )}{sin ^{2} ( 3 x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{1 - cos ( x ) + x sin ( x )}{3 sin ( 6 x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{x cos ( x ) + 2 sin ( x )}{18 cos ( 6 x )})

Plug in the value

x = 0

= \frac{0 \cdot cos ( 0 ) + 2 sin ( 0 )}{18 cos ( 6 \cdot 0 )}

Simplify

\frac{0 \cdot cos ( 0 ) + 2 sin ( 0 )}{18 cos ( 6 \cdot 0 )} : 0

= 0

What is the limit as x approaches 0 of ((x-xcos(x)))/(sin^2(3x)) ?
The limit as x approaches 0 of ((x-xcos(x)))/(sin^2(3x)) is 0