Question

Let F(x) = Sý sin*(0) dt. Evaluate the following limit . Let F(x) = $* sin?(t) dt. Evaluate the following limit. 2022

Answer 1

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a lim xso F(x) 2 int dt - x10 Jo lim x2 xso - So sink it dt = o form 10

i F(x) = dimr a rra 10x2 xrol de Jo 34 - S'entusu) 2 (t) det 0 2x because a (x") = nxn- Now by Leibnitz sale: - & pace nicado = /02) [4130) - fm)/ 3 ) – x floe)

8 fos e sindt dt] f(x)= 0 g(x)=x ah(t) = sind (t) - 3 Now h[g(x)] = h(x) [because g(x)= x =) h[glx)] = sin? (x) - [by putting tax inh(t)) and h[ f(x)) = ho] (because f(x) = o) - sin?(o) (by putting t=0 in h(t)) = (o) (because sin(o)=o] -) h[f(x)) = 0

(because sin(o)=o] lind Flet - o form again using Dili Hospital's sule: - - lingo Flx) = lim (dhe [sin (x)) d (2x) Vm x70 dX a to

- lim Um xto de (sin(x)) (2x) J da dim e simteshkasine)] because she (sin" (x) = n simma (eti ( sinta a (nx) =n because sin(x) cos(x) lim to d (sinx) = cosx x-o - lim fsin(x) cos(x) - sin(e) cos(o) because [ces (0) = 1) - lim F/x = 0 0(1) sin (0)=0 11 = 0