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Evaluate lim(x to oo) x("tan"^(-1)(x+1)/...

Evaluate `lim_(x to oo) x("tan"^(-1)(x+1)/(x+4)-(pi)/(4)).`

Text Solution

Verified by Experts

We have `underset(xtooo)limx("tan"^(-1)(x+1)/(x+4)-(pi)/(4))`
`=underset(xtooo)limx("tan"^(-1)(x+1)/(x+4)-tan^(-1)1)`
`=underset(xtooo)limxtan^(-1)(((x+1)/(x+4)-1)/(1+(x+1)/(x+4)))`
`=underset(xtooo)limxtan^(-1)((-3)/(2x+5))`
`=underset(xtooo)lim{(tan^(-1)((-3)/(2x+5)))/((-3)/(2x+5))}((-3)/(2x+5))`
`=underset(xtooo)lim{(tan^(-1)((-3)/(2x+5)))/((-3)/(2x+5))}underset(xtooo)lim((-3)/(2x+5))`
`=1xxunderset(xtooo)lim((-3)/(2+(5)/(x)))=1xx((-3)/(2))=-(3)/(2)`
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