>>768
R(δ)=max{|cosx-1| | |x|≦δ}とおいてδ>0にたいして
|∫[x=0,∞] cosx(n/({n^2}*{x^2}+1))dx-∫[x=0,∞] (n/({n^2}*{x^2}+1))dx|
≦|∫[x=0,∞] (cosx-1)(n/({n^2}*{x^2}+1))dx|
≦|∫[x=0,δ] (cosx-1)(n/({n^2}*{x^2}+1))dx|+|∫[x=δ,∞] (cosx-1)(n/({n^2}*{x^2}+1))dx|
≦R(δ)|∫[x=0,δ] (n/({n^2}*{x^2}+1))dx|+|∫[x=δ,∞] (cosx-1)(n/({n^2}*{x^2}+1))dx|
≦R(δ)|∫[x=0,∞] (n/({n^2}*{x^2}+1))dx|+|∫[x=δ,∞] (cosx-1)(n/({n^2}*{x^2}+1))dx|
≦R(δ)π/2+|∫[x=δ,∞] (cosx-1)(n/({n^2}*{x^2}+1))dx|
→R(δ)π/2 (n→∞)
なので
|lim(n→∞)∫[∫[x=0,∞] cosx(n/({n^2}*{x^2}+1))dx|-|lim(n→∞)∫[x=0,∞] (n/({n^2}*{x^2}+1))dx||≦R(δ)
が任意のδ>0についていえる。これを利用してできるとおもふ。