a=5 , b belongs to  R

by L Hôpitals rule,

after differentiating five times ,as the limit tends to zero the value tends to zero.so the value of a and b can be anything.

as xtends to 0,acosx +bxsinx -5)-->a-5 & denominator-->0 limit is finite if a-5=0i.e. a=5 so that it gets 0/0 form &limit can be found lim x-->0 (acosx + bxsinx -5)/x^5 (-asinx+b(xcosx+sinx))/5x^4 (-acosx+b(x(-sinx)+2cosx))/20x^3 =limx-->0 (-acosx-bxsinx+2bcosx)/20x^3 similarly (-a+2b)=0=>b=5/2