A function is defined in (0, $$\infty$$) by

$$f(x) = \left( {\matrix{ {1 - {x^2}} & {for} & {0 < x \le 1} \cr {\ln x} & {for} & {1 < x \le 2} \cr {\ln 2 - 1 + 0.5x} & {for} & {2 < x < \infty } \cr } } \right.$$

Which one of the following is correct in respect of the derivative of the function, i.e., f'(x) ?

If $$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\sin x} \over x} = l$$ and $$\mathop {\lim }\limits_{x \to \infty } {{\cos x} \over x} = m$$, then which one of the following is correct?

The left-hand derivative of f(x) = [x] sin ($$\pi$$x) at x = k, where k is an integer and [x] is the greatest integer function, is

The set of all points, where the function $$f(x) = \sqrt {1 - {e^{ - {x^2}}}} $$ is differentiable, is