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Thread: limit of multi variable functions

  1. #1
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    limit of multi variable functions

    So we were learning about 2 dimensional domain functions today, and my prof put up a limit. he did some work and then did something I was a bit stumped on for a while:  lim_{y\to\0} \frac{0}{y^2} \Rightarrow 0 . Isn't this indeterminate? Would it be 0 because y^2 is not ACTUALLY 0, but simple a number really close to 0 and then numerator is fixed at 0, so 0 divided by any number close to 0 is still 0. Am I right?

    P.S. How do you get  y \rightarrow 0 to go under the word "limit" rather than next to it?
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    Re: limit of multi variable functions

    I'll answer the easy part of your question.

    use \displaystyle

    \displaystyle{\lim_{y \to 0}} ~\dfrac {0}{y^2}

    $\displaystyle{\lim_{y \to 0}} ~\dfrac {0}{y^2}$

    Maybe one of the harder core math gurus here will chime in on the math of this.

    Note that with 2 applications of L'Hopital's rule you end up with

    $\displaystyle{\lim_{y \to 0}}~\dfrac 0 2 = 0$ in a form that's well defined.

    It's easy enough to see though that you can get $\dfrac 0 {y^2}$ arbitrarily close to zero (it's always zero) without y ever having to actually be zero thus making the quotient undefined.
    Last edited by romsek; Nov 9th 2015 at 12:56 PM.
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    Re: limit of multi variable functions

    Quote Originally Posted by toesockshoe View Post
    So we were learning about 2 dimensional domain functions today, and my prof put up a limit. he did some work and then did something I was a bit stumped on for a while:  \lim_{y\to\0} \frac{0}{y^2} \Rightarrow 0 . Isn't this indeterminate? Would it be 0 because y^2 is not ACTUALLY 0, but simple a number really close to 0 and then numerator is fixed at 0, so 0 divided by any number close to 0 is still 0. Am I right?

    P.S. How do you get  y \rightarrow 0 to go under the word "limit" rather than next to it?
    Look at the quote above the $y\to 0$ is correctly placed using \lim_{y\to\0} \frac{0}{y^2} , I added a \.

    You have ignored an important point in any \lim_{y\to a} f(y) it must be the case that $y\ne a$ always.

    If $y\ne 0$ then $\dfrac{0}{y^2}=0$ therefore the limit is $0$.
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    Re: limit of multi variable functions

    there appear to be differences in which LaTex interpreter you choose to use.


    \lim_{y \to 0} \dfrac 0 {y^2}

    $\lim_{y \to 0} \dfrac 0 {y^2}$

    \lim_{y \to \0} \dfrac 0 {y^2}

    $\lim_{y \to \0} \dfrac 0 {y^2}$

    $\displaystyle{\lim_{y \to 0}} \dfrac 0 {y^2}$
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    Re: limit of multi variable functions

    Quote Originally Posted by toesockshoe View Post
    Would it be 0 because y^2 is not ACTUALLY 0, but simple a number really close to 0 and then numerator is fixed at 0, so 0 divided by any number close to 0 is still 0. Am I right?
    Yes.
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