Let $$f\left( x \right) = x{\left( { - 1} \right)^{\left[ {\frac{1}{x}} \right]}},\,x \ne 0,$$      where $$\left[ x \right]$$ denotes the greatest integer less than or equal to $$x$$ then, $$\mathop {\lim }\limits_{x \to 0} f\left( x \right) = ?$$

Solution :
$$\eqalign{ & \because \,\left[ {\frac{1}{x}} \right] = {\text{ Integer}} \cr & \therefore \,{\left( { - 1} \right)^{\left[ {\frac{1}{x}} \right]}} = 1{\text{ or }} - 1 \cr & \mathop {\lim }\limits_{x \to 0} x{\left( { - 1} \right)^{\left[ {\frac{1}{x}} \right]}} \cr & = \mathop {\lim }\limits_{h \to 0} \left( h \right)\left( {1{\text{ or }} - 1} \right) = 0 \cr & = \mathop {\lim }\limits_{h \to 0} \left( { - h} \right)\left( {1{\text{ or }} - 1} \right) = 0 \cr} $$