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lim(xrarroo) (cot^(-1)(sqrt(x+1)+sqrtx))...

`lim_(xrarroo) (cot^(-1)(sqrt(x+1)+sqrtx))/(sec^(-1){((2x+1)/(x-1))^(x)})=`

A

1

B

`0`

C

`pi//2`

D

non existent

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(xrarroo)(lim)(sqrt(x+1)-sqrtx)=underset(xrarroo)(lim)(1)/(sqrt(x+1)+sqrtx)=0`
`rArr" "cot^(-1)(0)=pi//2`
Also,`" "underset(xrarroo)(lim)((2x+1)/(x-1))^(x)=underset(xrarroo)(lim)((2+(1)/(x))/(1-(1)/(x)))^(x)=oo`
`rArr" "sec^(-1)(oo)=pi//2`
Thus, required limit is 1.
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