Question

Evaluate $\underset{x\to 0}{lim}\frac{{\left(\cos x\right)}^{\frac{1}{2}}-{\left(\cos x\right)}^{\frac{1}{3}}}{{\sin }^{2}x}$.

Answer 1

$\underset{x\to 0}{lim}\frac{{\left(\cos x\right)}^{\frac{1}{2}}-{\left(\cos \right)}^{\frac{1}{3}}}{{\sin }^{2}x}\left(\frac{0}{0}form\right)$

Using L-Hospital rule

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

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

$=\frac{\frac{1}{3}-\frac{1}{2}}{2}$

$=\frac{2-3}{6\times 2}$

$=\frac{-1}{12}$