09/02/2007, 10:27 PM
jaydfox Wrote:"There is a ... such that for each ... there is a ...". Epsilon and h are going to zero, much as they would in limits, so why isn't there a quick shorthand as we use with limits?
If you can show it by the normal use of limits (for example \( \lim_{x\to a} f(x)\lim_{x\to a} g(x)=\lim_{x\to a} f(x)g(x) \)), its ok. But you know you can show everything with limits if you are not careful. And such thing like \( \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} > 0 \) is equivalent to \( \lim_{h\to 0} f(x+h) > f(x) \) is absolutely forbidden. And the \( \eps \),\( \delta \) definition is a clear way to specify what you mean.