Question

PLEASE ANSWER Q9 N Q10

5.2 Interchange of Limits and Integrals/ Derivatives 9) Calculate n cos lim -dr, 2n sin2 J2 justifying your calculation 10) Let 2nx fn(x) 1 +n2z2E0, 1], n > 1. Show that (fn) converges pointwise, but not uniformly, on [0,1] (b) Show that nevertheless, fa(a)d lim fa(a)dx. lim (This shows that the hypotheses of Theorem 5.8 are sufficient, but not nec- essary, for interchanging limits and integrals.)

Answer 1

Il ntcosa dr 2n +8ina fo(a)= n + cosx ant sina I'm fo(a) = f(a)= 1/2 I=(2,3] n-> 00 n+cosa goosa-sina olan + Bin²x) COME Intsinx I fala)- thao) = ons & I 3121431023) CE a Lantzina) 4€ = 1269n+Bin'as) =14h1+1 2 1 4h + asin?alls 3) ho = - ] ho depends on only & v exo, 3 no EN&t. fo(a) f(61)) LE & naho I= I'm In a no nyoo nooo of antsinza pont cosa da lim o 10 - ana xe[o] fr(s) = + n2a² so,a=o fla) = 0 limfo(a) = f(a) = (o, azon n> 00 nyo but folap = ett flo)= 0 So fo(a) does not convoges uniformly. low S for landas ndog [log114n?00) 7.lim bogenan?)..

I'lim folo) do = So fronda = Jo nyo