Derived small-angle limits for JEE

Formula

\[\lim_{x\to 0}\frac{\sin ax}{ax}=1,\quad \lim_{x\to 0}\frac{\sin ax}{x}=a,\quad \lim_{x\to 0}\frac{\tan ax}{x}=a,\quad \lim_{x\to 0}\frac{1-\cos ax}{x^2}=\frac{a^2}{2}\]

Variables: \(a\in\mathbb R\) constant.

Conditions: Radian measure.

Direct substitution after standard trigonometric limits.