Epsilon Delta Proof

[kidia]

Please can anybody help on this.

 

Use (beta,delta)definition of a limit to prove that

 

lim (x+y/x^2+y^2+1) = 0

(x,y)-(0,0)

[radiohead]

ugh, I get everything but the lim, which I am sure is pretty important. What kind of math is this?

[matt grime]

analysis, that's why it's in the analysis/calculus section.

 

small irony in kidia being answered by radiohead, just one letter....

 

anyway, the question, if i insert brackets, becomes find the limit of

 

[math]\frac{x+y}{x^2+y^2+1}[/math]

 

as (x,y) tends to (0,0).

 

call that fraction f(x,y), clearly |f(x,y)| < |x+y| since f(x,y) is just x+y divided by something bigger than 1 and if i take any positive Z, and divide by k where k>1 tehn Z/k< Z.

 

 

Now, suppose that |(x,y)| < d, ie (x,y) lies inside the circle of radius d centred on the origin in R^2. what is the largest value of |x| and the largest value of |y|? note i don't need to find the largest value of |x+y| since I can always overkill things in analysis; estimates are good enough if they work (slight tautology, sorry). but here

 

 

|f(x,y)|<|x+y| < |x|+|y| (triangle inequality) < something you should have found..

 

now, given e>0 pick d such that the "something you should have found" is less than e.

 

 

this is the standard way to do all e-d proofs:suppose the input to be less than d, use this to hopefully show that the ouptut is less than something dependent on d that can be made arbitrarily small, ie smaller then e.

[Pat Says]

We studied epsilon-delta proofs at the end of the year last year and I was kind of wondering if they have a purpose or are they just trivial? I could find a reason for most of the stuff we learned but not this one... even my teacher didn't like it.

[matt grime]

Trivial? Most poeple find them very difficult to begin with. The reason is to give a solid mathematical proof to the idea of continuity, better than the (incorrect) notion that sometinhg is continuous if you can draw its graph without taking your pen off the paper. Firstly, how do you know how to draw a graph ofs omething complicated? secondly this only applies to fucntions of one variable y=f(x), and do not tell us about the fucntions that we meet in the 'real world'. All it is is saying that we know y=f(x) (and these could be vectors now) and suppose we're allowed an error in measuring y (we always have errors) then how close to x must i make s so that f(s) is allowably close to y? The epsilon delta argument is just that set out properly.

 

OK, outside of a mthas course you're nto going to prove anything is or isnt' continuous, but you need to know what continuity means and what it implies. sadly a lot of teachers do not give the correct interpretation of continuity, why is a mystery. i generally explain it as a game. given f some function then we'll play a game. take x and f(x). you pick some e and if i can now pick a d so that for all y d within d of x f(y) is within e of f(x) then i win, otherwise you win. the points where i win f is continuous and if i win at all points x then f itself is continuous.

 

another way of saying the same thing is the f keeps things that a re close enough together close enough together.

 

 

example: define f(x) on the rationals, f=0 if x is lesss than sqrt(2) 1 if x is greater than sqrt(2)

 

this function is continuous even though it appears to have a jump at x=sqrt(2) but of course that is an irrational number.

 

in any case the best way to deal with continuity to begin with is that f is continuous at x if all sequences x_n tending to x satisfy f(x_n) tends to f(x)

[Pat Says]

So its uses aren't practical but the understanding of it is a necessity? I think I got it. that makes sense though, lots of things I learned earlier turned out to be useful in that way. Thank you for the explanation, it helped a lot