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Evaluate: ("lim")(n→oo)x[tan^(-1)((x+1)/...

Evaluate: `("lim")_(n→oo)x[tan^(-1)((x+1)/(x+2))-tan^(-1)(x/(x+2))]`

Text Solution

Verified by Experts

`underset(xtooo)limx["tan"^(-1)(x+1)/(x+2)-"tan"^(-1)(x)/(x+2)]`
`=underset(xtooo)limxtan^(-1)(((x+1)/(x+2)-(x)/(x+2))/(1+(x+1)/(x+2).(x)/(x+2)))`
`=underset(xtooo)limxtan^(-1)((x+2)/(2x^(2)+5x+4))`
`=underset(xtooo)lim(tan^(-)((x+2)/(2x^(2)+5x+4))/((x+2)/(2x^(2)+5x+4)))xx(x(x+2))/(2x^(2)+5x+4)`
`=1xx1/2=1/2`
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