Project ForgeEngineering Mathematics
Calculus
GATE 2022
Hard
The limit p = lim(x→π) [(x²+αx+2π²)/(x−π+2sin x)] has a finite value for a real α. The value of α and the corresponding limit p are
- α = −3π, and p = π
- Bα = −2π, and p = 2π
- Cα = π, and p = π
- Dα = 2π, and p = 3π
Explanation
Since the denominator → 0 as x→π, the numerator must too: π²+απ+2π² = 3π²+απ = 0, giving α=−3π. Applying L'Hôpital's rule with N'(x)=2x+α and D'(x)=1+2cos x: at x=π, N'(π)=2π−3π=−π and D'(π)=1+2(−1)=−1, so p=−π/−1=π.
Source: GATE ME 2022 Session 1 (ME-1) (2022)
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