Let $$f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.$$

Then at $$x=0$$

Let $$\lambda \in \mathbb{R}$$ and let the equation E be $$|x{|^2} - 2|x| + |\lambda - 3| = 0$$. Then the largest element in the set S = {$$x+\lambda:x$$ is an integer solution of E} is ______

The shortest distance between the lines $${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 6} \over 2}$$ and $${{x - 6} \over 3} = {{1 - y} \over 2} = {{z + 8} \over 0}$$ is equal to ________

The 4$$^\mathrm{th}$$ term of GP is 500 and its common ratio is $$\frac{1}{m},m\in\mathbb{N}$$. Let $$\mathrm{S_n}$$ denote the sum of the first n terms of this GP. If $$\mathrm{S_6 > S_5 + 1}$$ and $$\mathrm{S_7 < S_6 + \frac{1}{2}}$$, then the number of possible values of m is ___________