Q.Evaluate $\lim_{x \to \frac{\pi}{6}} \dfrac{\cot^2 x - 3}{\operatorname{cosec} x - 2}$.

At $x=\frac{\pi}{6}$, $\cot\frac{\pi}{6}=\sqrt{3}$ and $\operatorname{cosec}\frac{\pi}{6}=2$, so both numerator and denominator vanish — the form $\frac{0}{0}$. Use $\cot^2 x=\operatorname{cosec}^2 x-1$:

$\cot^2 x-3=\operatorname{cosec}^2 x-4=(\operatorname{cosec} x-2)(\operatorname{cosec} x+2).$ …

Step 1 — Check the form. At $x=\frac{\pi}{6}$, $\cot\frac{\pi}{6}=\sqrt{3}$ and $\operatorname{cosec}\frac{\pi}{6}=2$, so the numerator is $(\sqrt{3})^2-3=0$ and the denominator is $2-2=0$: the form $\frac{0}{0}$.

Step 2 — Rewrite the numerator. Using the identity $\cot^2 x=\operatorname{cosec}^2 x-1$,

Step 3 — Cancel the common factor. …