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 $\mathop {\lim }\limits_{x \to 0}\frac{sinx}{x}=1$)
e) $\mathop {\lim }\limits_{x \to 0}\frac{1-cos5x.cos7x}{sin^211x}$=$\mathop {\lim }\limits_{x \to 0}\frac{1- cos12x + 1 - cos 2x}{2sin^211x}$=$\mathop {\lim }\limits_{x \to 0}\frac{2cos^26x+2cos^2x}{2sin^211x}$=$\mathop {\lim }\limits_{x \to 0}\frac{(6x)^2.(\frac{cos6x}{6x})^2+x^2.(\frac{cosx}{x})^2}{(11x)^2.(\frac{sin11x}{11x})^2}$=$\mathop {\lim }\limits_{x \to 0}\frac{6.(\frac{cos6x}{6x})^2+(\frac{cosx}{x})^2}{11.(\frac{sin11x}{11x})^2}$= $\frac{6+1}{11}$=$\frac{7}{11}$ ( vì ta có $\mathop {\lim }\limits_{x \to 0}\frac{sinx}{x}=1$)
e) $\mathop {\lim }\limits_{x \to 0}\frac{1-cos5x.cos7x}{sin^211x}$=$\mathop {\lim }\limits_{x \to 0}\frac{1- cos12x + 1 - cos 2x}{2sin^211x}$=$\mathop {\lim }\limits_{x \to 0}\frac{2cos^26x+2cos^2x}{2sin^211x}$=$\mathop {\lim }\limits_{x \to 0}\frac{(6x)^2.(\frac{cos6x}{6x})^2+x^2.(\frac{cosx}{x})^2}{(11x)^2.(\frac{sin11x}{11x})^2}$=$\mathop {\lim }\limits_{x \to 0}\frac{
36.(\frac{cos6x}{6x})^2+(\frac{cosx}{x})^2}{1
21.(\frac{sin11x}{11x})^2}$= $\frac{
36+1}{1
21}$=$\frac{
37}{1
21}$ ( vì ta có $\mathop {\lim }\limits_{x \to 0}\frac{sinx}{x}=1$)