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lim(xto oo) (int(0)^(x)tan^(-1)t\ dt)/(s...

`lim_(xto oo) (int_(0)^(x)tan^(-1)t\ dt)/(sqrt(x^(2)+1))` is equal to

A

`(pi)/2`

B

`(pi)/4`

C

1

D

`pi`

Text Solution

Verified by Experts

The correct Answer is:
A
`lim_(xto oo) (int_(0)^(x)tan^(-1)tdt)/(sqrt(x^(2)+1))` (`(oo)/(oo)` form)
`=lim_(xto oo) (tan^(-1))/(x/(sqrt(x^(2)+1)))`
`=(pi)/2 lim_(xto oo) (sqrt(x^(2)+1))/x`
`=(pi)/2 lim_(xto oo) sqrt(1+1/(x^(2)))=(pi)/2`
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