For each ositive integer n, define a fun...

For each ositive integer n, define a function `f _(n) on [0,1]` as follows:
`f _(n((x)={{:(0, if , x =0),(sin ""(pi)/(2n), if , 0 lt x le 1/n),( sin ""(2pi)/(2n) , if , 1/n lt x le 2/n), (sin ""(3pi)/(2pi), if, 2/n lt x le 3/n), (sin "'(npi)/(2pi) , if, (n-1)/(n) lt x le 1):}` Then the value of `lim _(x to oo)int _(0)^(1) f_(n) (x) dx ` is:

`pi`

`pi/2`

`1/pi`

`2/pi`

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