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Let f: RvecR be a differentiable functio...

Let `f: RvecR` be a differentiable function having `f(2)=6,f^(prime)(2)=1/(48)dot` Then evaluate `("lim")_(xvec2)int_6^(f(x))(4t^3)/(x-2)dt`

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`lim_(xto2)int_(6)^(f(x))(4t^(3))/(x-2)dtlim_(xto2)(int_(6)^(f(x))4t^(3)dt)/(x-2)[0/0 "form"]`
`=lim_(xto2)(4(f(x))^(3)f'(x))/1`
(applying L' Hopital rule)
`=4(f(2))^(3)xxf'(2)`
`=4(6)^(3)xx1/48`
`=18`
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