Evaluate the following limit:
`lim_(nto oo)[1/(n^(2))"sec"^(2)1/(n^(2))+2/(n^(2))"sec"^(2)4/(n^(2))+…………+1/nsec^(2)1]`

Options

A. `(1)/(2) tan1`
B. tan 1
C. `(1)/(2) "cosec"1`
D. `(1)/(2) sec1`

Answer

A

Explanation

We have
`underset(n to oo) lim{(1)/(n^2)"sec"^(2)(1)/(n^(2))+(2)/(n^(2))+"sec"^(2)(4)/(n^(2))+....+(1)/(n)"sec"^(2)(n^(2))/(n^(2))}`
`underset(n to oo) lim underset(r=1)overset(n)sum(r)/(n^(2))"sec"^(2)(r^(2))/(n^(n))`
`underset(n to oo) lim underset(r=1)overset(n)sum{(r)/(n)"sec"^(2)((r)/(n))^(2)}(1)/(n)`
`-underset(0)overset(1)int x sec^(2) x^(2)dx=-(1)/(2)[tan x^(2)]_(0)^(1)=(1)/(2)tan`