Let \( \mathrm{f}:(0, \infty) \rightarrow(0, \infty) \) be a differentiable function such that \( f(1)=e \) and \( \lim …

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

[JEE Main 2020, 4 Sep (Shift 2)]