There is still another Method to solve the Problem.
We subst. #x=tan^2y rArr" As "x to 0, y to 0.#
#:." The Reqd. Lim.="lim_(x to 0) ((xsqrtx-1)/sqrt(x+1)+1)/x,#
#=lim_(y to 0){(tan^2ysqrt(tan^2y)-1)/sqrt(tan^2y+1)+1}/tan^2y,#
#=lim_(y to 0)((tan^3y-1)/secy+1)/tan^2y,#
#=lim_(y to 0){(sin^3y-cos^3y)/(cos^3y*secy)+1}/(sin^2y/cos^2y),#
#=lim_(y to 0){(sin^3y-cos^3y)/cos^2y+1}(cos^2y/sin^2y),#
#=lim_(y to 0)(sin^3y-cos^3y+cos^2y)/cos^2yxxcos^2y/sin^2y,#
#=lim_(y to 0) (sin^3y-cos^3y+cos^2y)/sin^2y,#
#=lim_(y to 0) sin^3y/sin^2y+(cos^2y-cos^3y)/sin^2y,#
#=lim_(y to 0) siny+cos^2y/sin^2y*(1-cosy),#
#=lim_(y to 0) siny+cos^2y/sin^2y*((1-cosy)(1+cosy))/(1+cosy),#
#=lim_(y to 0) siny+cos^2y/sin^2y*sin^2y/(1+cosy),#
#=lim_(y to 0) siny+cos^2y/(1+cosy),#
#=sin0+cos^2 0/(1+cos0),#
#=0+1^2/(1+1).#
# rArr" The Reqd. Lim.="1/2,# as before!
Enjoy Maths.!