Question

Evaluate: $\underset{x\to 0}{lim}\frac{si{n}^{2}x}{\sqrt{3+cosx}-2}$

Answer 1

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

$=\underset{x\to 0}{lim}\frac{(1-{\cos }^{2}x)(\sqrt{3+\cos x}+2)}{3+\cos x-4}$

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

$=\underset{x\to 0}{lim}-(1+\cos x).(\sqrt{3+\cos x}+2)$

$=-8$