Question

Evaluate $\underset{x\to 0}{lim}\frac{(1-cosx\sqrt{cos2x})}{x^2}$

Answer 1

$\underset{x\to 0}{lim}\frac{(1-cosx\sqrt{cos2x})}{x^2}$

$=\underset{x\to 0}{lim}\{\frac{(1-cosx\sqrt{cos2x})}{x^2}\times \frac{(1+cosx\sqrt{cos2x})}{(1+cosx\sqrt{cos2x})}\}$

$=\underset{x\to 0}{lim}\frac{(1-cosxcos2x)}{x^2}\times \frac{1}{(1+cosx\sqrt{cos2x})}$

$=\underset{x\to 0}{lim}\frac{(1-cosxcos2x)}{x^2}\times =\underset{x\to 0}{lim}\frac{1}{(1+cosx\sqrt{cos2x})}$

$=\underset{x\to 0}{lim}\left\{\frac{1-co{s}^{2}x(1-2si{n}^{2}xco{s}^{2}x)}{x^2}\right\}\times \frac{1}{(1+11\sqrt{1})}$

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

$=\frac{1}{2}\times \underset{x\to 0}{lim}\frac{(1-cos{}^{2}x+2sin{}^{2}xco{s}^{2}x)}{x^2}\times \frac{1}{2}\times \underset{x\to 0}{lim}\frac{s{n}^{2}x(1+2co{s}^{2}x)}{x^2}$

$=\frac{1}{2}\times \underset{x\to 0}{lim}{\left(\frac{sinx}{x}\right)}^2\times \underset{x\to 0}{lim}(1+2co{s}^{2}x)$

$=\frac{1}{2}\times 1^2\times (1+2\times 1^2)=\frac{3}{2}$