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Find the limit as x goes to 0 of (tan (2x)) / (sin (3x))

Evaluate the limit.

    \[ \lim_{x \to 0} \frac{\tan (2x)}{\sin (3x)}. \]


Using the previous exercise we know

    \[ \lim_{x \to 0} \frac{\sin (ax)}{\sin (bx)} = \frac{a}{b}. \]

Therefore, we can evaluate this limit as follows,

    \begin{align*}  \lim_{x \to 0} \frac{\tan (2x)}{\sin (3x)} &= \lim_{x \to 0} \left( \frac{1}{\cos (2x)} \cdot \frac{\sin (2x)}{\sin (3x)} \right) \\[9pt]  &= 1 \cdot \frac{2}{3} = \frac{2}{3}. \end{align*}

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