JEE Main 2025 — 29 January, Morning Shift — Question 67

$\lim _{x \rightarrow 0^{+}}\left(x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\ldots . .+\left[\frac{p}{x}\right]\right)-x^{2}\left(\left[\frac{1}{x^{2}}\right]+\left[\frac{2^{2}}{x^{2}}\right]+ . . +\left[\frac{9^{2}}{x^{2}}\right]\right)\right) \geq 1$

$(1+2+\ldots \ldots+p)-\left(1^{2}+2^{2}+\ldots +9^{2}\right) \geq 1$ $\frac{\mathrm{p}(\mathrm{p}+1)}{2}-\frac{9.10 .19}{6} \geq 1$

$\mathrm{p}(\mathrm{p}+1) \geq 572$

Least natural value of $p$ is 24