inkyvoyd Group Title Applying the epsilon-delta limit definition one year ago one year ago

1. inkyvoyd Group Title

I'll just state wikipedia's definition for my own reference. $$\forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon)$$

2. electrokid Group Title

ok

3. inkyvoyd Group Title

the example my textbook gives is the following: "Show that the $$\lim_{x\rightarrow1}(5x-3)=2$$"

4. inkyvoyd Group Title

Then, they use substitutions with what is given c=1,L=2.

5. inkyvoyd Group Title

So we show this statement to be true? $$0<|x-1|<\delta \rightarrow|f(x)-2|<\epsilon$$

6. electrokid Group Title

yes.. the wiki definition can be elaborately expanded as below: $\text{if:} \lim_{x\to a^-}f(x)=\lim_{x\to a+}f(x)\\ \text{then,}\lim_{x\to a}f(x)\quad\text{exists and}\\ \lim_{x\to a^-}f(x)=\lim_{x\to a+}f(x)=\lim_{x\to a}f(x)$

7. electrokid Group Title

and $\epsilon\to0$

8. electrokid Group Title

as $$x\to1$$, $$\delta\to0\implies\epsilon\to0$$

9. inkyvoyd Group Title

okay. Then the book says "We find delta by working backwards from the epsilon-inequality: $$|(5x-3)-2|=|5x-5|<\epsilon$$ $$|x-1|<\epsilon/5$$

10. inkyvoyd Group Title

so they essentially manipulate the f(x)-L expression to arrive at the x-c expression?

11. electrokid Group Title

that would be backward mapping... map "f(x)" to "x" given "L"

12. electrokid Group Title

called image

13. electrokid Group Title

it is very similar to the process of obtaining the vertical asymptote of a function.

14. inkyvoyd Group Title

What I don' get is the last part. "thus we can take $$\delta=\epsilon/5$$ If 0<|x-1|<delta=epsilon/5, then $$|(5x-3)-2|=5|x-1|<5(\epsilon/5)=\epsilon$$

15. electrokid Group Title

so $$\epsilon$$ exists and hence the limit is true. Also, could you tell me WHY they made that particular scalar multiple for $$\delta$$?

16. electrokid Group Title

ooooh

17. inkyvoyd Group Title

does that 5 give us any hints as to the rate the limit converges? or does it just pop up conincidentally?

18. electrokid Group Title

$0<|x-1|<\delta\\ \epsilon>|(5x-3)-2|=5|x-1|<5\delta\\$ now, for a certain "x", $\epsilon=5\delta$

19. electrokid Group Title

now, what you said about hint of "derivatives" is not wrong.. Leibnitz's approach in declaring $\lim_{\delta\to0}{\epsilon\over\delta}={dy\over dx}$ was not a fluke... many stalwarts did not realize that connection till the former dared to state it.

20. inkyvoyd Group Title

hmm- this is a bit offtopic, but why is it that one can often treat a differential (and differential notation) as actual numbers, but they don't follow all rules? I heard somewhere that one can redefine differentials as another set of numbers, but there are rules that they dont' follow that real numbers do follow.

21. electrokid Group Title

in fact, if you search into the "Antique stronghold" of "Project Gutenberg" and "Google Books", you will stumble into the rare Math bibles. (I was very upset on the concept of eigen values and I found the history of development of matrices on Gutenberg)

22. electrokid Group Title

I do not understand the last question

23. electrokid Group Title

why dont you give it a shot for f(x)=x^2-1 in the above process... you'd end up weith the "First principle of differentiation" :)

24. inkyvoyd Group Title

well, leibniz interpreted dy/dx as an infinitisimal change of y divide by an infinitismal change in x. You can do things with differential form like solving separable differential equations such as dy/dx=1/x you rewrite in differential form to get dy=1/x dx or y=ln x (differential form rewriting is like treating the notation as a fraction?) also, you could just directly integrate Integral dy/dx dx=integral 1/x dx integral dy=ln x y=ln x The notation is weird because it suggests that dy and dx are atual numbers, but everyone says they aren't. what should I do with f(x)=x^2-1?

25. electrokid Group Title

no, they are not. they are "rational" functions

26. electrokid Group Title

you can play with rational numbers as long as you avoid "divisions with "0""

27. inkyvoyd Group Title

yes, but doesn't dx approach zero? I mean, youg et the indeterminant form 0/0 in all derivatives when evalating the difference quotient limit with direct substitution...

28. electrokid Group Title

"approach", yes, "equal it?" no you will still have some infinitesimally small $$\delta$$, but never "0" to avoid these problems, you use L'Hopital's Rule....

29. inkyvoyd Group Title

mm- I'ma go ahead and try some practice with this epsilon delta stuff- my main motivation is to use it for riemann sums, but I'll have to understand this first. Thanks for all the help!

30. electrokid Group Title

surething

31. electrokid Group Title

nice day.. started at 6am with an incredible integration problem here ending with a good talk on limits.

32. inkyvoyd Group Title

xD was it the integral of sqrt(tan x) by any chance?

33. electrokid Group Title

dont remember exactly.. it was a form of Beta function

34. inkyvoyd Group Title

oh wow... I got a while before I'll even remotely understand that...

35. electrokid Group Title

i htink i was$\int_0^1\frac{\mathrm{d}x}{\sqrt{1-x^m}}$