## Babynini one year ago Proving limits with delta..

1. Babynini

2. Babynini

Gaah sorry it is sideways. I got stuck at the end there, not sure where to go. The answer, I think, should be some min{ ?}

3. Babynini

@jim_thompson5910 you free? :)

4. zepdrix

Let delta be sufficiently small, say $$\large\rm \delta<1$$. Then, $$\large\rm |x-2|<1$$$\large\rm -1< x-2<1$Adding 6,$\large\rm 5<\color{royalblue}{x+4<7}$And I think we can do something withhhhh this. Hmm. I'm just trying to figure out how we relate this to absolute value.$\large\rm |x-2||x+4|<|x-2|\cdot 7<\epsilon$

5. zepdrix

$\large\rm |x-2|<\frac{\epsilon}{7}$And then you have to choose your delta carefully, depending on which is bigger, this epsilon/7 or the 1 restriction that we put on delta.$\large\rm \delta=\min\left\{1,~~\frac{\epsilon}{7}\right\}$

6. zepdrix

Yes, we were trying to find a way to "deal with" the |x+4| :)

7. zepdrix

I was able to get this relationship: $$\large\rm 5<\color{royalblue}{x+4<7}$$ But not something like this: $$\large\rm a< |x+4|<b$$ I hope that isn't going to matter.. I'm trying to think about it :\ hmm

8. Babynini

hrmm what you did looks pretty close to what we did in class with other examples.

9. jim_thompson5910

@zepdrix I think as long as a > 0 and b > 0, then $\Large a < x+4 < b \implies |a| < |x+4| < |b|$

10. Babynini

So then it comes out like: ...?

11. zepdrix

For part $$\large\rm i)$$ at the last line I think you want to write:$\large\rm |x-2||x+4|<\epsilon$I don't think we can relate this to delta just yet at that point. hmm.

12. zepdrix

And then in part $$\large\rm ii)$$ I would maybe make a connection here:$\large\rm 5<x+4<7\qquad\implies\qquad |x+4|<7$

13. Babynini

ahh right right.

14. zepdrix

And after that "So" line I would get rid of the equals sign on the next line. It reads like, the epsilon from the previous line is equivalent to the next stuff. Maybe just the word "Then," instead of "="

15. zepdrix

Oh also, maybe too much words at the start of $$\large\rm ii)$$ hehe Let $$\large\rm \delta$$ be $$\large\rm \delta<1$$. When I'm reading this in my head it reads like: "Let delta be delta is less than 1."

16. zepdrix

Assume $$\large\rm \delta<1$$. Maybe that sounds a little nicer >.< I dunno

17. Babynini

Dude I was just looking at that too. So redundant haha

18. zepdrix

These are tricky huh? :d it's like combing Math and English class! lol I'm taking a whole class on these types of problems :3 it's a pain sometimes.

19. Babynini

Oh goshh, really? So you're pro then!

20. Babynini

hahaha best description of these ever. But I like English, so it's ok xP

21. Babynini

Is it all looking good now? :)

22. zepdrix

Nooo I'm not a pro :O noob. learning. Looking good? I think soooo.

23. zepdrix

We don't want a therefore on the last line. We gotta pick a delta first! :D Use another word like ummm... Let delta=min{stuff}. Therefore, 0<|x-2|<delta implies |f(x)-L|<epsilon or some nonsense like that, ya? :D

24. zepdrix

Maybe I'm being a little fussy :) But these words mean things

25. Babynini

Ah well I think you're doing excellently then! xP oh gosh. What is all this fanciness haha. Nah I think it's good. Fussy is better than having it incomplete!

26. zepdrix

Are you near the beginning of a Calc 1 course? :o That's when I remember first going over these types of problems. Everything was super fun after this stuff. Related rates, max/min problems, mean-value theorem, lots of cool normal math stuff.

27. Babynini

mhm! I am taking Calc 1 this quarter.

28. Babynini

Okay! one last time!! xD

29. Babynini

oh goood, I hope it gets more fun haha

30. zepdrix

I don't understand the first line of your proof: $$\large\rm \delta=\qquad\qquad >0$$ Do you go back later and fill in that gap after you have found a delta or something?

31. Babynini

oh, yes. That is how the prof does it. So now i'd go back and put in the delta = min{1, e/7} in there. I just forgot to do that yet xD

32. Babynini

I guess I could just write delta > 0 there though. Since I go back at the end and do that fancy "Therefore ..."

33. zepdrix

Hmm that's a clever way to do that :) I like that. Yah either way should be ok I think, since you're stating your delta that you found at the end.

34. Babynini

Would it be redundant/dumb to do both? o.o

35. zepdrix

You mean, would it be dumb to put $$\large\rm \delta=\left\{1,\frac{\epsilon}{7}\right\}>0$$ at the top since you already have it at the bottom? Nahhh seems fine. Why are they numbered i) and ii)? Is that just a way to stay organized?

36. Babynini

Ah ok. Yep, it just helps me from getting lost haha

37. zepdrix

woah woah woah we're forgetting something very crucial

38. zepdrix

wow im glad i caught this before it was too late

39. zepdrix

$\Huge \blacksquare$

40. zepdrix

buhahaha

41. Babynini

hahaha I will put it in right now!!

42. Babynini

thanks!

43. zepdrix

yay team