Limits, Continuity & Differentiability
Limits
nta_pyq_2025_jan
Grade 12
Question:
Let [t] be the greatest integer less than or equal to t. Then the least value of p \in N for which lim x\to0 + (x ([ x1 ] + [ x2 ] + \ldots + [ xp ]) - x 2 ([ x12 ] + [ x2 2 ] + \ldots + [ x9 2 ])) \ge 1 is equal 2 2 to ________. x
Step-by-Step Solution
Key Concept: Apply the core result for standard limits and expansions and simplify using the given constraints.
2 2 1 2 p 1 2 9 2 lim (x ([ ] + [ ] + \ldots..+[ ]) - x ([ ] + [ ] + [ ])) \ge 1 (24) x\to0 + x x x x 2 x 2 x 2 2 2 2 (1 + 2 + \ldots \ldots + p) - (1 + 2 + \ldots9 ) \ge 1 p(p + 1) 9.10.19 - \ge 1 2 6 p(p + 1) \ge 572 Least natural value of p is 24
Correct Answer: 24