Let $f:[0,1] \rightarrow \mathbb{R}$ and $g:[0,1] \rightarrow \mathbb{R}$ be defined as follows :

$\left.\begin{array}{rl}f(x) & =1 \text { if } x \text { is rational } \\ & =0 \text { if } x \text { is irrational }\end{array}\right]$ and

$\left.\begin{array}{rl}g(x) & =0 \text { if } x \text { is rational } \\ & =1 \text { if } x \text { is irrational }\end{array}\right]$ then

$$\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\sqrt n } \over {\sqrt {{n^3}} }} + {{\sqrt n } \over {\sqrt {{{(n + 4)}^3}} }} + {{\sqrt n } \over {\sqrt {{{(n + 8)}^3}} }} + .... + {{\sqrt n } \over {\sqrt {{{[n + 4(n - 1)]}^3}} }}} \right\}$$ is

Let $$f(x) = {1 \over 3}x\sin x - (1 - \cos \,x)$$. The smallest positive integer k such that $$\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^k}}} \ne 0$$ is

Let $$f:[1,3] \to R$$ be a continuous function that is differentiable in (1, 3) an

f'(x) = | f(x) |² + 4 for all x$$ \in $$ (1, 3). Then,