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Math Help - Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume

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    Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume

    second to last question.

    Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume that L is finite.
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    Quote Originally Posted by tn11631 View Post
    second to last question.

    Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume that L is finite.
    That's pretty straight forward. In terms of the definitions of those limits you want to prove that
    "Given \epsilon> 0 there exist \delta> 0 such that if |x-x_0|< \delta then |f(x)- L|< \epsilon"

    if and only if
    "Given \epsilon> 0 there exist \delta> 0 such that if |y|< \delta then |f(x_0+ y)- L|< \epsilon.
    (I've changed "x" to "y" in the second part so as not to confuse the two uses of "x".)

    Comparing those two, they will be the same if x_0+ y= x. In other words, let y= x- x_0.
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    In other words, let y= x- x_0.....do i have to show anything more or do the two definitions take of it? Sorry i'm really bad at proofs..
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tn11631 View Post
    In other words, let y= x- x_0.....do i have to show anything more or do the two definitions take of it? Sorry i'm really bad at proofs..
    Do exactly as HallsOfIvy said. The basic idea is that in the limit \lim_{x\to x-0}f(x)=0 if we let z=x-x_0 then we get \lim_{z\to 0}f(z)=0. Formalize this.
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