e) $\mathop {\lim }\limits_{x \to 0}\frac{1-cos5x.cos7x}{sin^211x}$
=$\mathop {\lim }\limits_{x \to 0}\frac{1- cos12x + 1 - cos 2x}{2sin^211x}$
=$\mathop {\lim }\limits_{x \to 0}\frac{2cos^26x+2cos^2x}{2sin^211x}$
=$\mathop {\lim }\limits_{x \to 0}\frac{(6x)^2.(\frac{cos6x}{6x})^2+x^2.(\frac{cosx}{x})^2}{(11x)^2.(\frac{sin11x}{11x})^2}$
=$\mathop {\lim }\limits_{x \to 0}\frac{36.(\frac{cos6x}{6x})^2+(\frac{cosx}{x})^2}{121.(\frac{sin11x}{11x})^2}$
= $\frac{36+1}{121}$=$\frac{37}{121}$ ( vì ta có $\mathop {\lim }\limits_{x \to 0}\frac{sinx}{x}=1$)