Check the continuity

Question

Explain for continuity the function f defined by f(x) = \(lim\ n→ ∞ \frac{e^x - x^nsin(nx)}{1 + x^n}\) (0≤x≤π/2) at x = 1. Explain why the function ƒ does not vanish any where in [0, π/2] although f(0)f(π/2) <0.