Zadanie 9b, Paper 1, listopad 2023, TZ1, Matematyka AA IB HL

Write $f(x)=[\sin(kx)/x]^2/(1+x^2)$. The inner quotient is 0/0; L’Hôpital gives limit kcos(kx)/1→k. The remaining factor tends to 1, hence lim f=k². Thus k²=25 and k=5 because k>0. Alternatively, sin(kx)=kx+O(x³), so numerator =k²x²+O(x⁴); dividing gives k²+O(x²) and the same limit.

M1:factor the squared quotient and continuous rational factor, or begin a valid series/twice-L’Hôpital method. A1:correct resulting form. M1:evaluate the removable sine quotient by L’Hôpital, standard sine limit with scaling, or series. A1:quotient limit k, or correct leading numerator k²x². A1:full limit k² including the factor tending to 1. A1:k=5, excluding −5. For direct twice-L’Hôpital: M1 recognize 0/0; M1 differentiate numerator and denominator; A1 for [2k sin(kx)cos(kx)]/[2x+4x³]; M1 differentiate again; A1 limit k²; A1 k=5. For a direct series route: M1 attempt the sine series; A1 leading squared numerator k²x²; M1 substitute into the quotient; independent R1 for reducing the limit to k² plus vanishing terms; A1 limit k²; A1 k=5. Missing explicit limit symbols alone are not penalized.