math question

2. (18 Points) Let f(x) = x2 + 1 (3 - cost + e-) and F(x) = f (t) dt. (a) (4 Points) Evaluate lim f(x). I-+ +00 (b) (4 Points) Determine whether + 00 f(x) dx converges or not. (c) (10 Points) Given that f'(x) - ex (1-x-x2 -23) + (-3x2 + (x2 + 1)xsinc+ (x2 -1) cosx +3) (x2 + 1)2 f"(x) = e-* (-2-5x + 4x3 + 2x4 + x5) + (-2(x4 -1) sina+ (x4 + 7)rcosr + 6x3 -18.r) (x2 + 1)3 Find the quadratic Taylor polynomial of F at the reference point 0 and use it to approximate F(0.1). In addition, derive an upper bound for the absolute error committed