The exercises in my imaginary textbook are giving me an ε, say .001, & are making me find a delta, such that all values of x fall within that ε range of .001. The section that I'm working on is called "proving limits." Well, that is not proving a limit. All that's doing is finding values of what f(x) could be by whatever ε was given, in our case .001. Any reasonable person could understand that by making ε smaller, the values of what f(x) could be, are gonna be closer to the limit, L. But still that isn't proving the limit. So if I can't make ε out to be any small discrete value, how am I supposed to prove a limit? Well, if you could prove that delta was a function of ε, then this would work. Why? Because you won't have to put yourself in that position → you don't actually have to pick an ε, so you won't be stuck with just a range of possible f(x)'s. Take, for example, f(x) = 5x. Given ε>0, d=ε/5 will satisfy. d=ε/5 says that for ANY ε given (smaller than any number you can think of), delta will always be the ε given divided by 5. The fraction is irrelevant. It's simply the fact that delta is a function of ε that proves the limit.(adsbygoogle = window.adsbygoogle || []).push({});

Here's a new one, & a relevant one to what differential calculus is supposed to be about. Suppose f(x)=x². DQ reads: 2x+h. lim x² h→0 =2x. I understand the whole 0/0 thing & the need for a limit, but I don't understand how the delta ε def'n works here. First of all, how am I supposed to graph this? In the previous example where f(x)=5x, it was graphed as such & ε was the range of f(x)'s around the limit point, L, & the deltas were simply some unknown range around x. Pretty straight forward. Well how does that work in our new example? I think I'm going to stop here. Delta is no longer just an arbitrary range around x. It's an arbitrary range around x+h. And ε is of course, the range of f(x)'s around the limit point L, but in this case, the output, f(x), is going to be the slope over the interval x+h. Am I supposed to be graphing h as a function of 2x+h? That doesn't even make sense...

:/

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# B How does the delta ε definition prove derivatives?

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