If function $f\left(x\right)$ is differentiable at $x=a$ then $\underset{x\to a}{lim} \frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$ is

 For $x\ne a,\frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$

$\begin{array}{l}=\frac{\left(x^2-a^2\right)f\left(a\right)+a^2\left(f\right(a)-f(x)}{x-a}\\ =(x+a)f\left(a\right)-a^2\frac{f\left(x\right)-f\left(a\right)}{x-a}\end{array}$

Therefore,   $\underset{x\to a}{lim} \frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}=2af\left(a\right)-a^2{f}^{\mathrm{\prime }}\left(a\right)$