Question

lim (x tends to infinity).
(x+2)arctan(x+2) -x(arctan(x))

Answer 1

lim(x--->infty) {(x+2)arctan(x+2)-x.arctanx}

this is in the form of infty - nifty
 use expansion of arctanP = pi/2 - 1/P + 1/3P^3 - 1/5P^5+........infty
then,
  lim(x--->infty)[(x+2){pi/2-1/(x+2)+1/3(x+2)^3-1/5(x+2)^5....infty}-x{pi/2-1/x+1/3x^3-1/5x^5+.....infty} ]
 =lim(x--->infty)[pi/2(x+2)-1+1/3(x+2)^2-1/5(x+2)^4..... -pi/2x +1 -1/3x^2+1/5x^4.....]
 put x--->infty 
= 0 -1 + 0 -0......-0+1-0+0-0......
=0 

hence answer is 0