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Thread: Limit of a constant

  1. #1
    Senior Member BAdhi's Avatar
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    Limit of a constant

    if $\displaystyle f(x)=c$ where c is a constant and it is asked to find

    $\displaystyle \lim_{x \to a} f(x) = L$

    According to the definition we can take

    $\displaystyle \forall \epsilon >0 , \exists\delta>0$

    $\displaystyle 0<|x-a|<\delta \implies |f(x)-L| < \epsilon$

    my question is whether $\displaystyle \delta$ and $\displaystyle \epsilon$ are independent from each other
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  2. #2
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    Quote Originally Posted by BAdhi View Post
    if $\displaystyle f(x)=c$ where c is a constant and it is asked to find

    $\displaystyle \lim_{x \to a} f(x) = L$

    According to the definition we can take

    $\displaystyle \forall \epsilon >0 , \exists\delta>0$

    $\displaystyle 0<|x-a|<\delta \implies |f(x)-L| < \epsilon$

    my question is whether $\displaystyle \delta$ and $\displaystyle \epsilon$ are independent from each other

    Of course not: $\displaystyle \delta$ almost always depends of $\displaystyle \epsilon$ (and on the function and on what point is x tending

    to, of course), but in this case it's trivially true that $\displaystyle L=c\,,\,\,\delta=$ whatever , as you can easily check.

    Tonio
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  3. #3
    MHF Contributor

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    The largest possible value of $\displaystyle \delta$ depends on $\displaystyle \epsilon$ but $\displaystyle \delta$ can be any number smaller than that largest possible value. For this particular problem, the largest possible value of $\displaystyle \delta$ is infinity (there is no largest possible value).
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