# Solve for: \lim_{x arrow 0} ((tan(2x)-x)/(3x-sin(x)))

## Expression: $\lim_{x \rightarrow 0} \left(\frac{ \tan\left({2x}\right)-x }{ 3x-\sin\left({x}\right) }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow 0} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \tan\left({2x}\right)-x \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 3x-\sin\left({x}\right) \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ 2{\sec\left({2x}\right)}^{2}-1 }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 3x-\sin\left({x}\right) \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ 2{\sec\left({2x}\right)}^{2}-1 }{ 3-\cos\left({x}\right) }\right)$

Evaluate the limit

$\frac{ 2{\sec\left({2 \times 0}\right)}^{2}-1 }{ 3-\cos\left({0}\right) }$

Simplify the expression

\begin{align*}&\frac{ 1 }{ 2 } \\&\begin{array} { l }0.5,& {2}^{-1}\end{array}\end{align*}

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