For x∈(−π2,π2), if y(x)=∫cosecx+sinxcosecxsecx+tanxsin2xdx and limx→(π2)−y(x)=0 then y(π4) is equal to

For x∈(−π2,π2), if y(x)=∫cosecx+sinxcosecxsecx+tanxsin2xdx and limx→(π2)−y(x)=0 then y(π4) is equal to

Let f:[−π2,π2]→R be a differentiable function such that f(0)=12, If the limx→0x∫x0f(t)dtex2−1=α,then 8α2 is equal to :

Let f:→R→(0,∞) be strictly increasing function such that limx→∞f(7x)f(x)=1. Then, the value of limx→∞[f(5x)f(x)−1] is equal to

Statement I: The equation (sin−1x)3+ (cos−1x)3−aπ3=0 has a solution for all a≥132.Statement II: For any x∈R,sin−1x+cos−1x=π2 and 0≤(sin−1x−π4)2≤9π216

Suppose f(x)𝑓(𝑥) and g(x)𝑔(𝑥) are equal for all x-values except x=t𝑥=𝑡. If limx→tf(x)=Llim𝑥→𝑡⁡𝑓(𝑥)=𝐿, then is limx→tg(x)=Llim𝑥→𝑡⁡𝑔(𝑥)=𝐿 true?