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Math Help - Basic limits help

  1. #1
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    Basic limits help

    Hiya, I'm having to learn about limits really quickly and am struggling to fully grasp the concept. While I understand the basic concept, I'm having trouble figuring out what can be considered as (dis-)proof of a limit, and also the Epsilon-Delta definition of a limit. I can't seem to find anything that offers an explaination of these which is not in mathematical terms - I am not a mathematician, hence the difficulty.

    I have a few example functions and would appreciate any help, although I would prefer more descriptive help rather than just the maths. All functions are defined as 0 for (x,y)=(0,0) and I would like to test the continuity at (0,0).

    1. f(x,y)=\frac{xy^3}{x^2+y^6}

    2. f(x,y)=\frac{xy}{x^2+xy+y^2}

    3. f(x,y)=\frac{x^2+y^2}{x^2-y^2}

    Here are my approaches to these functions...

    Function 1.

    We can rewrite the function as f(x,y)=\frac{\sqrt{x^2}}{\sqrt{x^2+y^6}}*\frac{\sq  rt{y^6}}{\sqrt{x^2+y^6}}. My notes feature fractions with the same powers for both numerator and denominator - is this proof of a limit?

    Function 2.

    If we consider the result of f(x,y) along the line f(x)=y, the function becomes f(x,y)=\frac{x^2}{x^2+x^2+x^2} = \frac{1}{3}. As this is constant, but the function is defined as (0,0) at (x,y)=(0,0), we have a large step rather than a gradual convergence, therefore f(x,y) is not continuous at (0,0).

    Function 3.

    If we consider the function when x=0 and y=1, the result of the function, the result will be -1 and 1 respectively. As with function 2, we have a constant value along these lines everywhere except (0,0), therefore f(x,y) is not continuous at (0,0).

    Thank you in advance for any help or advice you can offer.
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  2. #2
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    Here's a site that explains the concept of limits with multivariable functions very nicely, with examples.

    Pauls Online Notes : Calculus III - Limits
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  3. #3
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    Thanks for the link, lots of clear descriptions for many topics aswell as for limits. I'm now fairly confident in my answers for functions 2+3, but realise that the limit does not exist in function 1 as it resolves to \frac{1}{2} when x=y^3. Could anyone confirm this?
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  4. #4
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    Quote Originally Posted by CorruptioN View Post
    Thanks for the link, lots of clear descriptions for many topics aswell as for limits. I'm now fairly confident in my answers for functions 2+3, but realise that the limit does not exist in function 1 as it resolves to \frac{1}{2} when x=y^3. Could anyone confirm this?
    I get all your answers to be correct. Though for problem 2 I think you meant to say approach along the line y=x, not f(x)=y.
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  5. #5
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    Yup, that's what I meant. Thanks for your help!
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