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Epsilon-Delta proofs

  1. Sep 11, 2007 #1
    Hello. I have an upcoming exam for my math course and I am aware that much of it will revolve around Epsilon-Delta proofs. My understanding of them is good enough to prove most limits, but I would be more comfortable being able to answer anything that is thrown at me on this test :confused:.

    I would appreciate example proofs and recommendations of books that would help on this subject.

    Thanks :biggrin:.

    EDIT: To specify, it is a multi variable calculus course so I'm studying multi variable Epsilon-Delta proofs.
    Last edited: Sep 11, 2007
  2. jcsd
  3. Sep 12, 2007 #2
    come on guys, I really need some help with this.
  4. Sep 12, 2007 #3
    Give an epsilon-delta proof for the existence of the limit,
    \lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} \frac{(x^2)-(y^2)}{x+y}
  5. Sep 12, 2007 #4
    I know that x^2-y^2 is always less than x^2+y^2, but I cannot find a relation with x+y. If I simplify, I cannot find a relation between sqrt(x^2+y^2) and x+y.

    EDIT: sorry. I meant to say x+y
    Last edited: Sep 12, 2007
  6. Sep 12, 2007 #5


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    Are you aware that [itex]x^2- y^2= (x-y)(x+y)[/itex]? Using that it is easy to see that
    [tex]\frac{x^2- y^2}{x- y}= x+ y[/itex] (as long as x, y are not both 0). you don't need a relation with x-y, but one between [itex]\sqrt{x^2+ y^2}[/itex] and x+ y.

    Since [itex]\delta[/itex] is the "distance from the point" or, in this case, from 0, it often helps to convert to polar coordinates. [itex]x= r cos(\theta)[/itex] and [itex]y= r sin(\theta)[/itex] so [itex]x^2- y^2= r^2cos^2(\theta)- r^2 sin^(theta)= r^2(cos^2(\theta) - sin^2(\theta)[/itex] while x- y= r cos(\theta)- r sin(\theta)= r(cos(\theta)- sin(\theta))= r(cos(\theta)- sin(\theta)) so
    [tex]\frac{x^2- y^2}{x-y}= r \frac{cos^2(\theta)- sin^(\theta)}{cos(\theta)- sin(\theta}[/tex].
    That can be simplified but the crucial point is that r multiplying the fraction. You can take "r" to be "[itex]\delta[/itex]" and argue that for any angle [itex]\theta[/itex] you can make that quantity as small as you please by making r small.
  7. Sep 12, 2007 #6
    Thx. I know that the polar conversions work real well, but I never really got the hang of them.
  8. Sep 13, 2007 #7
    lol, then i suggest you get the hang of them before your test... they are gonna keep coming up in this class, it’s multivariable calculus right?
  9. Sep 13, 2007 #8
    Ya. Its multivariable differential calculus. Its kinda weird how many proofs that are put in this class. A teacher at my siblings school says that multivariable calculus is not this proof oriented. Epsilon-Delta proofs are involved, and that it understandable, but the textbook is not even remotely close to the material on the test as the material on the test is mostly proving the formulas from the book.
  10. Sep 13, 2007 #9
    Maths is always proof orientated... if it's not, it's not maths, just calculation.
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