Question

$$p(x) = \begin{cases} \lim_{n \to \infty} \left( \lambda + \frac{p(x)}{n} \sum_{r=1}^{n} \frac{r^2 - e^{-x} + r - 1}{r(r+1)} \right), & x > 0 \\ p, & x = 0 \\ \lim_{n \to \infty} \sum_{r=1}^{n} \frac{r^2 + r + e^{x} - 1}{r(r+1)}, & x < 0 \end{cases}$$

Now, match the following columns

Column-I (A) Number of points of dis-continuity of $f(x) = ||x|| - ||x||$ in [-3,100] is equal to (B) The reciprocal of the sum of values of k for which $f(x) = (k + |x|)^2 e^{(5-|x|)^2}$ is differentiable for all real x, is equal to (C) If f is a differentiable function satisfying $f'(0) = 2, f(0) = 0, f(x + y^2) = f(x) + y f(y) \forall x, y \in R$, then $f'(2) + f(3)$ equals (D) The value of $p'(\ln 2) + p'(\ln \frac{1}{2}) + p'(\ln \frac{3}{2^2}) + p'(\ln \frac{5}{2^3}) + \dots \infty$

Answer 1

A → Q, B → S, C → R, D → P

Answer 2

The problem is fundamentally flawed due to inconsistencies in the definition of p(x) and the resulting values not matching the options.