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lim(n->oo)1/n[tan (pi/(4n))+tan((2 pi)/(...

`lim_(n->oo)1/n[tan (pi/(4n))+tan((2 pi)/(4n))+...tan((npi)/(4n))]`

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`lim_(x->o) (tan x)/n = 1`
`lim_(x-> oo) [ (tan (pi/(4n)))/n xx (pi/(4n))/((pi)/(4n)) + (tan((2 pi)/(4n)))/n xx ((2 pi)/(4n))/((2 pi)/(4n)) + ........... + ( tan((n pi)/(4n)))/n] xx((n pi)/(4n))/(( n pi)/(4 n))`
`lim_(x-> oo) [ pi/(4 n^2) + ( 2 pi)/(4 n^2) + ( 3 pi)/(4 n^2) + ...... + (n pi)/(4n^2)]`
`lim_(x-> oo) (pi xx (1+2+3+...+n))/(4 n^2)`
`lim_(x-> oo) ( (pi xx (n xx(n+1)))/2)/(4n^2)`
`= pi/8 lim_(x->oo) (n^2(1+1/n))/(n^2)`
`= pi/8`
Answer
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