Home
Class 12
MATHS
Evaluate: ("lim")(nvecoo)[1/(n^2)sec^2 1...

Evaluate: `("lim")_(nvecoo)[1/(n^2)sec^2 1/(n^2)+2//n^2sec^2 4/(n^2)++1/nsec^2 1]`

Text Solution

Verified by Experts

The correct Answer is:
`1/2 tan1`
`lim_(n to oo) [1/(n^(2)) "sec"^(2)1/(n^(2))+2/(n^(2))"sec"^(2) 4/(n^(2))+……..+1/nsec^(2)1]`
`=lim_(nto oo) 1/n sum_(r=1)^(n)r/n"sec"^(2)(r/n)^(2)`
`=int_(0)^(1)x sec^(2) x^(2) dx`
Put `x^(2)=t` so that `2x dx=dt`
When `x=0, t=0`, When `x=1, t=1`
`:. `Required limit `=1/2 int_(0)^(1) sec^(2) t dt `
`=1/2 [tan t]_(0)^(1)=1/2[tan 1-0]`
`=1/2 tan 1`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.3|4 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.4|10 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.1|4 Videos

  • CURVE TRACING

    CENGAGE|Exercise Exercise|24 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Evaluate: ("lim")_(ntooo)(n^p sin^2 (n !))/(n+1)

Evaluate ("lim")_(nvecoo)sum_(k=1)^nk/(n^2+k^2)

Evaluate: ("lim")_(nvecoo)((a1 1/x+a2 1/x++a n1/x)/n)^(n x)

Evaluate: ("lim")_(nvecoo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))++1/(sqrt(3n^2)))

Evaluate: ("lim")_(nvecoo)n^(-n^2){(n+2^0)(n+2^(-1))(n+2^(-2))(n+2^(-n+1))}^n

Evaluate: lim_(n->oo)(((n+1)(n+2)...(n+n))^(1/n))/n

Evaluate lim_(ntooo) [(1+2+3+...+n)/(5n^2+2n+1)]

Evaluate lim_(ntooo) [(1+2+3+...+n)/(4n^2-3n+2)]

Evaluate: lim_(n->oo)[1/(n a)+1/(n a+1)+1/(n a+2)++1/(n b)]

Evaluate lim_(ntooo) (1)/(1+n^(2))+(2)/(2+n^(2))+...+(n)/(n+n^(2)).