f:(0,∞)→(0,∞) be a differentiable function such that f(1)=e and limt→xt2f2(x)-x2f2(t)t-x=0 If f(x)=1, then x is equal to:

f:(0,∞)→(0,∞)  f(1)=elimt→xt2f2(x)-x2f2(t)t-x Using L' hospitals rule limt→x2tf2(x)-x2·2f(t)f'(t)1=2xf2(x)-x22f(x)f'(x)=0⇒2xf(x)=2x2f'(x)f'(x)f(x)=1x Integrate lnf(x)=lnx+Cf(1)=e⇒C=1lnf(x)=lnx+1f(x)=ex If f(x)=1=ex⇒x=1e