Question

(a) Compute lim 1+0 Jo ecos 1 de. You must justify the passage of the limit under the integral sign. .x2 + (b) Suppose {vn) is a sequence of real numbers that converges to yo. Define F(y) = (sin(y) dar. Prove that Fyn) + F(v.). (i.e. F is a continuous function of y.)

Answer 1

li m e cosn dn fn Cmy= é^cosn ', Kae [0,m) tm fr Er) = 0 is Zerco. ei mit funetiog poituise +mt Lo) e(tn my) of atfainy mestimm at min by uping Jnd dircirative tegf,

* By manr Evet Foremly om [aw). e cosn limm4Ya) Ox boavdadl finetion I dm Ane. trat $%3- mehere thof Conneregey to yo. y o be a Grinen rreal leq ket Flo)= (^ Stnl ym) I+m? up. Hm?

Now Sin(m) s a bounded funetron on [oro0) anf Itmz Boundey Arut iun on [+). Sin( hm) zontimouy an t is a funet pm of beandwal on [Oco). napjation is Wni Poml y 1+m2 [Ocw). continuory "linl lim Fltn): lim 1+m2 up. Sinlym) Itm? a contimoy Bnufion at y=Jo. * FY) is a continmony Fumetiog f became tis arbitepey.