Calculate `lim_(x->oo) 1/x^2 int_0^x f(t) dt` `f(x)=cos x -1+ (.5x^2)`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the limit of definite integral such that:

`lim_(x->oo) (1/x^2)* int_0^x f(t) dt`

Since the problem provides the equation of the function, then you need to substitute `(cos t - 1 + 0.5t^2)`  for `f(t)`  such that:

`int_0^x f(t) dt = int_0^x(cos t - 1 + 0.5t^2) dt`

`int_0^x f(t) dt = int_0^x cos t dt - int_0^x dt + 0.5int_0^x t^2 dt`

`int_0^x f(t) dt = sin t|_0^x - t|_0^x + 0.5t^3/3|_0^x`

`int_0^x f(t) dt = sin x - sin 0 - x + 0 + 0.5x^3/3 - 0.5*0^3/3`

`int_0^x f(t) dt = sin x - x + 0.5x^3/3`

Hence, evaluating the limit yields:

`lim_(x->oo) (1/x^2)* int_0^x f(t) dt = lim_(x->oo) (1/x^2)*(sin x - x + 0.5x^3/3)`

`lim_(x->oo) (1/x^2)* int_0^x f(t) dt = lim_(x->oo) sin x/x^2 - lim_(x->oo) x/x^2 + 0.5/3lim_(x->oo) x^3/x^2`

`lim_(x->oo) (1/x^2)* int_0^x f(t) dt = 1*0 + 0 + 0.5/3*oo`

`lim_(x->oo) (1/x^2)* int_0^x f(t) dt = oo`

Hence, evaluating the given limit under the given conditions yields `lim_(x->oo) (1/x^2)* int_0^x f(t) dt = oo` .

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