A function is defined as follows

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

Which one of the following is correct in respect of the above function?

Consider the following statements

1. $$x + {x^2}$$ is continuous at x = 0

2. $$x + \cos {1 \over x}$$ is discontinuous at x = 0

3. $${x^2} + \cos {1 \over x}$$ is continuous at x = 0

Which of the above are correct?

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?