$$\mathop {\lim }\limits_{x \to 0} \left[ {{\text{cose}}{{\text{c}}^3}x \cdot \cot \,x - 2\,{{\cot }^3}x \cdot {\text{cosec}}\,x + \frac{{{{\cot }^4}x}}{{\sec \,x}}} \right]$$         is equal to :

Solution :
$$\eqalign{ & \mathop {\lim }\limits_{h \to 0} \left[ {{\text{cose}}{{\text{c}}^3}x \cdot \cot \,x - 2\,{{\cot }^3}x \cdot {\text{cosec}}\,x + \frac{{{{\cot }^4}x}}{{\sec \,x}}} \right] \cr & = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\cos \,x}}{{{{\sin }^4}x}} - \frac{{2\,{{\cos }^3}x}}{{{{\sin }^4}x}} + \frac{{{{\cos }^5}x}}{{{{\sin }^4}x}}} \right) \cr & = \mathop {\lim }\limits_{x \to 0} \frac{{\cos \,x{{\left( {1 - {{\cos }^2}x} \right)}^2}}}{{{{\sin }^4}x}} \cr & = \mathop {\lim }\limits_{x \to 0} \cos \,x \cr & = 1 \cr} $$