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)