Home » Blog » Evaluate the given limit

Evaluate the given limit


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

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.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):