x->0
tanx = x+(1/3)x^3 +o(x^3)
sinx = x-(1/6)x^3 +o(x^3)
tanx -sinx = (1/2)x^3 +o(x^3)
lim(x->0) [√(1+tanx) -√(sinx+1) ]/[xln(1+x^2) ]
=lim(x->0) [√(1+tanx) -√(sinx+1) ]/x^3
=lim(x->0) [(1+tanx) -(sinx+1) ]/{ x^3 .[√(1+tanx) + √(sinx+1) ] }
=lim(x->0) (tanx -sinx)/{ x^3 .[√(1+tanx) + √(sinx+1) ] }
=lim(x->0) (1/2)x^3/{ x^3 .[√(1+tanx) + √(sinx+1) ] }
=lim(x->0) (1/2)/[√(1+tanx) + √(sinx+1) ]
=(1/2) / 2
=1/4