Let f x = 1 + cos x π − x 2 ⋅ sin 2 x ln 1 + π 2 − 2 π x + x 2   x ≠ π λ   x = π is continuous at x = π , then λ is equal to

Step-by-step solution

For f x to be continuous at x = π , lim x → π ⁡ f x = f π Now, lim x → π ⁡ 1 + cos ⁡ x π - x 2 · sin 2 ⁡ x ln ⁡ 1 + π - x 2 = λ Putting x = π + t , we get, ∴ lim t → 0 1 − cos t t 2 ⋅ sin 2 t ln 1 + t 2 = λ lim t → 0 1 2 ⋅ sin 2 t t 2 ⋅ t 2 ln 1 + t 2 = λ 1 2 × 1 × 1 = λ ⇒ λ = 1 2