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Let f: RvecR be a differentiable functio...

Let `f: RvecR` be a differentiable function such that `f(0),f(pi/2)=3a n df^(prime)(0)=1.` If `g(x)=int_x^(pi/2)[f^(prime)(t)cos e ct-cottcos e ctf(t)]dtforx(0,pi/2],` then `(lim)_(xvec0)g(x)=`

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`f(0),f(n/2),f'(0)=1`
`g(x)=int_x^(pi/2)[f'(t)cosect=cottcosectf(t)]dt`
`=int_x^(pi/2)f'(t)cosectdt-int_x^(pi/2)f(t)cott*cosectdt`
`=[f(t)cosect]_x^(pi/2)+int_x^(pi/2)f(t)cosectcottdt-int_x^(pi/2)f(t)cott*cosectdt`
`g(x)=[f(pi/2)cosec(pi/2)-f(x)cosecx]=3-f(x)cosecx`
`g(x)=3-f(x)cosecx`
`lim_(x->0)g(x)=lim_(x-->0)8-f(x)cosecx`
`=3-lim_(x-->0)f(x)/sinx=3-lim_(x-->0)(f'(x))/cosx`
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