1. ## Limit theorem

I would appreciate some tips on this.

2. $\lim_{x\to x_0}\frac{xf(x_0)-x_0f(x)}{x-x_0}=\lim_{x\to x_0}\frac{xf(x_0)-x_0f(x_0)+x_0f(x_0)-x_0f(x)}{x-x_0}=$

$=\lim_{x\to x_0}\frac{f(x_0)(x-x_0)}{x-x_0}-\lim{x\to x_0}\frac{x_0(f(x)-f(x_0)}{x-x_0}=$

$=\lim_{x\to x_0}f(x_0)-x_0\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=f(x_0)-x_0f'(x_0)$