If $f(x)$ is a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim \limits_{t \to x} \large\frac{t^2f(x)-x^2f(t)}{t-x} $$=1$ fopr each $x >0$ then $f( \large\frac{3}{2})$ is equal to :

Question

$\begin{array}{1 1} (1) \large\frac{13}{6} \\ (2) \large\frac{23}{18} \\ (3) \large\frac{25}{9} \\ (4) \large\frac{31}{18} \end{array} $