Question

Find the value of $\underset{x\to 0}{lim}\frac{x^3\cot x}{1-\cos x}$.

Answer 1

$\underset{x\to 0}{lim}\frac{x^3\cot x}{1-\cos x}$

$\underset{x\to 0}{lim}\frac{x^3\cot x}{1-\cos x}\times \frac{1+\cos x}{1-\cos x}$

$=\underset{x\to 0}{lim}\frac{x^3\cot x(1+\cos x)}{1-{\cos }^{2}x}=\underset{x\to 0}{lim}\frac{x^3\cot x(1+\cos x)}{{\sin }^{2}x}$

$=\underset{x\to 0}{lim}\frac{x^3\cos x(1+\cos x)}{\sin x{\sin }^{2}x}=\underset{x\to 0}{lim}\frac{x^3\cos x(1+\cos x)}{{\sin }^{3}x}$

$=\underset{x\to 0}{lim}\left(\frac{x^3}{{\sin }^{3}x}\right)\times \underset{x\to 0}{lim}\cos x\times \underset{x\to 0}{lim}(1+\cos x)=\underset{x\to 0}{lim}\left[\frac{1}{{\left(\frac{\sin x}{x}\right)}^3}\right]\times \cos 0\times (1+\cos x)$

$=\frac{1}{1}\times 1\times (1+1)$

$=2$