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## Homework Statement

I tried to solve this limit, but failed in the process 'cause the answer's 1 but I get 0. I'd just like for you to check out my steps and tell me what I've done wrong, not post a different solution. Thanks in advance.

[tex]\mathop {\lim }\limits_{x \to 0} \frac{{2\sin \left( x \right) - \sin \left( {2x} \right)}}{{{x^3}}}[/tex]

## Homework Equations

[tex]\mathop {\lim }\limits_{x \to 0} \frac{{\sin \left( x \right)}}{x} = 1[/tex]

## The Attempt at a Solution

[tex]\begin{array}{l}

\mathop {\lim }\limits_{x \to 0} \frac{{2\sin \left( x \right) - \sin \left( {2x} \right)}}{{{x^3}}} = \mathop {\lim }\limits_{x \to 0} \frac{{2\sin \left( x \right)}}{{{x^3}}} - \frac{{\sin \left( {2x} \right)}}{{{x^3}}} = \mathop {\lim }\limits_{x \to 0} \left( 2 \right)\left( {\frac{{\sin \left( x \right)}}{x}} \right)\left( {\frac{1}{{{x^2}}}} \right) - \left( {\frac{{\sin \left( {2x} \right)}}{{2x}}} \right)\left( {\frac{2}{{{x^2}}}} \right) \\

= \mathop {\lim }\limits_{x \to 0} \left( 2 \right)\left( 1 \right)\left( {\frac{1}{{{x^2}}}} \right) - \left( 1 \right)\left( {\frac{2}{{{x^2}}}} \right) = \mathop {\lim }\limits_{x \to 0} \frac{2}{{{x^2}}} - \frac{2}{{{x^2}}} = 0 \\

\end{array}[/tex]