Limits, Continuity & Differentiability
Limits
nta_pyq_2025_jan
Grade 12
Question:
lim x\to0 cosec x (\sqrt2 cos 2 x + 3 cos x - \sqrtcos 2 x + sin x + 4) is:
0
1 \sqrt15
1 2\sqrt5
- 1 2\sqrt5 x
Step-by-Step Solution
Key Concept: Apply the core result for standard limits and expansions and simplify using the given constraints.
lim cosec(\sqrt2 cos 2 x + 3 cos x - \sqrtcos 2 x + sin x + 4) x\to0 (4) 2 cos ec x (cos x + 3 cos x - sin x - 4) lim x\to0 2 2 (\sqrt2 cos x + 3 cos x + \sqrtcos x + sin x + 4) 2 1 (cos x + 3 cos x - 4) - sin x lim x\to0 sin x 2 2 (\sqrt2 cos x + 3 cos x + \sqrtcos x + sin x + 4) x (cos +4) (cos x - 1) - sin x lim x\to0 2 2 sin x (\sqrt2 cos x + 3 cos x + \sqrtcos x + sin x + 4) 2 x x x -2 sin (cos x + 4) - 2 sin cos 2 2 2 lim x\to0 x x 2 2 2 sin cos (\sqrt2 cos x + 3 cos x + \sqrtcos x + sin x + 4) 2 2 x x - (sin (cos x + 4) + cos ) 2 lim x\to0 x 2 2 cos (\sqrt2 cos x + 3 cos x + \sqrtcos x + sin x + 4) 2 1 - 2\sqrt 5 x
Correct Answer: 4