Alternate Definition of a Derivative Question

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The alternate form of the derivative is

$$f'(a)=\lim_\limits{x \to a}\frac{f(x)-f(a)}{x-a}$$

Can this be rewritten to this and still be true?

$$f'(a)=\lim_\limits{x \to a}\frac{f(a)-f(x)}{a-x}$$

If what I wrote violates any limit rule please tell me.

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1 Answer

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Yes. But you're not using any properties of limits. Notice that the fractions inside the limits are actually the same:

$$\dfrac{f(x)-f(a)}{x-a} = \dfrac{-(-f(x)+f(a))}{-(-x+a)} = \dfrac{-1}{-1} \cdot \dfrac{f(a)-f(x)}{a-x} = \dfrac{f(a)-f(x)}{a-x}$$

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