So I added a picture of the problem I am working on. I can't begin to prove this because I don't even understand why it is true. It seems so trivial but can someone explain this?Attachment 30380

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- Mar 11th 2014, 08:20 PMrkushner83Real Analysis: Derivatives and Limits of Functions
So I added a picture of the problem I am working on. I can't begin to prove this because I don't even understand why it is true. It seems so trivial but can someone explain this?Attachment 30380

- Mar 11th 2014, 08:57 PMromsekRe: Real Analysis: Derivatives and Limits of Functions
start with a form of the definition of a derivative at a point a

$f'(a) = \displaystyle{\lim_{x \to a}}\dfrac{f(x)-f(a)}{x-a}$

for the if direction assume there exists $\epsilon(x)$ that goes to 0 as $x \to a$ and show this leads to the above limit

for the only if direction assume the derivative exists, i.e. the above limit exists, and show the statement leads to requiring the limit of $\epsilon(x)$ to be as stated.