If differentiable, then it's continuous theorem proof;

$$\large\lim_{x \rightarrow x_0}f(x)-f(x_0)=0$$
So from Prof. David Jerison's explanation, we can write this as
$$\large\lim_{x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)$$
my question is, how did he go from this, to
$$\large\lim_{x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)=f'(x_0)*0=0$$?

Question

If differentiable, then it's continuous theorem proof;

$$\large\lim_{x \rightarrow x_0}f(x)-f(x_0)=0$$
So from Prof. David Jerison's explanation, we can write this as
$$\large\lim_{x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)$$
my question is, how did he go from this, to
$$\large\lim_{x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)=f'(x_0)*0=0$$?

anonymous · Answer

According to the second definition of derivative,
f'(x(0)) = lim (as x approaches x(0)) [f(x) - f(x(0)]/[x - x(0)]
Further as x --> x(0), x - x(0) = 0.

zepp · Answer

Um, I see, so it's all about the definition of derivatives, thanks :)

anonymous · Answer

Yes, just the definition of derivative, but the one that most students don't remember.