Let f:(0, ∞) →(0, ∞) be a differentiable function such that f(1)= e and lim_t → x (t^2 f^2(x)-x^2 f^2( t ))/(t -x)=0. If f(x)=1, then x is equal to:

Let $\displaystyle f:(0, \infty) \rightarrow(0, \infty)$ be a differentiable function such that $\displaystyle f(1)=\mathrm{e}$ and $$\lim _{\mathrm{t} \rightarrow x} \frac{\mathrm{t}^2 f^2(x)-x^2 f^2(\mathrm{t})}{\mathrm{t}-x}=0 . $$ If $\displaystyle f(x)=1$, then $\displaystyle x$ is equal to: