Let b be a nonzero real number. Suppose f : R $$ \to $$ R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation $$f'(x) = {{f(x)} \over {{b^2} + {x^2}}}$$

for all x$$ \in $$R, then which of the following statements is/are TRUE?

Which of the following inequalities is/are TRUE?

If $$I = \sum\nolimits_{k = 1}^{98} {\int_k^{k + 1} {{{k + 1} \over {x(x + 1)}}} dx} $$, then

Let
$$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \over 4}} \right)....\left( {{x^2} + {{{n^2}} \over {{n^2}}}} \right)}}} \right)^{{x \over n}}},$$ for

all $$x>0.$$ Then