IV- (8 points) Part A Let g be the function defined over 10; +[ by g(x) = x²-21nx. 1) Calculate lim g(x) and lim g(x). 0+-x x>0 X-+00 2) Set up the table of variations of g. 3) Prove that g(x)>0 over 10; +[. Part B 2(1+Inx) Let f be the function defined over 10; +[ by f(x)=x-2+ and designate by (C) its X representative curve in the orthonormal (O;i; j). 1) Calculate lim f(x) and lim f(x). Deduce an asymptote to (C). x0 x>0 2) a)Show that the straight line (d) of equation y=x-2 is an asymptote to(C) at +xo. b)Study, according to the values of x, the relative positions of (C) and (d). 3) Prove that f'(x) = g(x) then set up the table of variations of f. 4) Show that the equation f(x) = 0 admits a unique root α, then verify that 0.54 < a <0.56. 5) Determine the coordinates of the point B on(C) where the tangent (T) at B is parallel to (d). 6) Show that f admits an inverse function f' and determine its domain of definition. 7) Calculate (f¹)'(0) in terms of a. 8) Draw (d) and (C). 9) Calculate the area of the domain limited by (C), (d) and the two lines x =1 and x = e.