Question

Evaluate: ${lim}_{x\to 0}\frac{\cos x-\cot x}{x}$

Answer 1

$\underset{x\to 0}{lim}\frac{\cos x-\cot x}{x}$

Apply L-Hospital Rule

$\underset{x\to 0}{lim}\frac{-\sin x+{\text{cosec}}^{2}x}{1}=\underset{x\to 0}{lim}\frac{1-{\sin }^{3}x}{{\sin }^{2}x}$

Apply L-Hospital Rule

$\underset{x\to 0}{lim}\frac{-3{\sin }^{2}x\cos x}{2\sin x\cos x}=\underset{x\to 0}{lim}-\frac{3}{2}\sin x=0$