In a calculus book I found:
Evaluate the following limits in terms of the number $\alpha =\lim_{x \to 0} \frac{\sin x}{x}$
i) $$ \lim_{x \to 0} \frac{\sin 2x}{x} $$
I know some techniques how to evaluate limits, but I don’t understand what it means to evaluate a limit in terms of another number alpha.
Can someone give an example of what’s expected here?
Hint: write it as $\;\displaystyle\lim_{x \to 0} \frac{\sin 2x}{x} = \color{red}2 \cdot \lim_{x \to 0} \frac{\sin 2x}{\color{red}{2}x} = 2 \,\cdot\, \underbrace{ \lim_{y \to 0} \frac{\sin y}{y}}_{\alpha}\,$.