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OpenStudy (zepp):
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 \[\large\lim_{x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)=f'(x_0)*0=0\]?
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OpenStudy (anonymous):
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.
OpenStudy (zepp):
Um, I see, so it's all about the definition of derivatives, thanks :)
OpenStudy (anonymous):
Yes, just the definition of derivative, but the one that most students don't remember.
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