Question

${lim}_{x\to 0}\frac{sin{x}^2(1-cos{x}^2)}{x^6}$

Answer 1

${lim}_{x\to 0}\frac{sin{x}^2(1-cos{x}^2)}{x^6}$

$={lim}_{x\to 0}\frac{sin{x}^2\times 2si{n}^2\frac{x^2}{2}}{x^6}$

$={lim}_{x\to 0}\frac{sin{x}^2}{x^2}\times {lim}_{x\to 0}\frac{2si{n}^2\frac{x^2}{2}}{x^4}$

$=1\times 2\times {\left({lim}_{x\to 0}\frac{sin\frac{x^2}{2}}{\frac{x^2}{2}}\right)}^2\times \frac{1}{4}$

$=\left(1\right)\times 2\times 1\times \frac{1}{4}[\because {lim}_{x\to 0}\frac{sinx}{x}=1]$

$=\frac{1}{2}$