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Evaluate the given limit

by RoRi

Compute

    \[ \lim_{t \to 0} \frac{\sin (\tan t)}{\sin t}. \]

First, we recall \lim_{x \to 0} \frac{\sin x}{x} = 1. So, we would like to rewrite the given function in a form that looks something like this. So,

    \begin{align*} \lim_{t \to 0} \frac{\sin (\tan t)}{\sin t} &= \lim_{t \to 0} \left( \frac{1}{\cos t} \cdot \frac{\sin (\tan t)}{\sin t / \cos t} \right)\\ &= \lim_{t \to 0} \left( \frac{1}{\cos t} \cdot \frac{\sin (\tan t)}{\tan t} \right)\\ &= \left(\lim_{t \to 0} \frac{1}{\cos t} \right) \left( \lim_{t \to 0} \frac{\sin (\tan t)}{\tan t} \right)\\ &= 1 \cdot 1 \\ &= 1. \end{align*}

Where we used that \lim_{t \to 0} \tan t = 0 and so \lim_{t \to 0} \frac{\sin (\tan t)}{\tan t} = 1.

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