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lim(n->oo)1/n[1/(n+1)+2/(n+2)+....+(3 n)...

`lim_(n->oo)1/n[1/(n+1)+2/(n+2)+....+(3 n)/(4 n)]`

A

log4

B

`-log4`

C

`1-log4`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
D
`underset(nrarroo)(lim)(1)/(n)((1)/(n+1)+(2)/(n+2)+...+(3n)/(4n))`
`=underset(nrarroo)(lim)(1)/(n)(((1)/(n))/(1+(1)/(n))+((2)/(n))/(1+(2)/(n))+...+((3n)/(n))/(1+(3n)/(n)))`
`=underset(nrarroo)(lim)(1)/(n)sum_(r=1)^(3n)(((r)/(n))/(1+(r)/(n)))=int_(0)^(3)(x)/(1+x)dx`
`=int_(0)^(3)(x+1-1)/((1+x))dx=int_(0)^(3)dx-int_(0)^(3)(1)/(1+x)dx`
`=[x-log(x+1)]_(0)^(3)=3-log4`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-MHT CET Corner
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