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Math Help - help proving lim question

  1. #1
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    help proving lim question

    hello you all, i'm new here.. i'd like if you help me in this question.

    f and g are function defined in a punctured neighborhood of X0, and L is a real number.
    assuming lim(x->x0) (f times g)(x)=L

    i need to prove\disprove that if lim(x->x0) f(x)=infinity, then lim(x->x0) g(x) exists.


    it would be very helpful if you also instruct me in the general way of proving such questions, because i'm kinda new in this subject...
    thx!
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  2. #2
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    Re: help proving lim question

    Are you allowed to take for granted that the limit of a product is equal to the product of limits?
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  3. #3
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    Re: help proving lim question

    Quote Originally Posted by Prove It View Post
    Are you allowed to take for granted that the limit of a product is equal to the product of limits?
    sorry, but i didn't understand you..
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  4. #4
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    Re: help proving lim question

    Prove It was asking if you have the theorem that says "if lim_{x\to a} f(x)= F and \lim_{x\to a}g(x)= G then \lim_{x\to a} f(x)g(x)= FG. However, that does not appear to me to apply here since \lim_{x\to x_0} f(x) does not exist. Instead, it should be apparent that, in order that \lim_{x\to x_0} f(x)g(x) exist, the limit of g(x) must not only exist but must be a very specific number!
    Last edited by HallsofIvy; March 27th 2013 at 07:01 AM.
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  5. #5
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    Re: help proving lim question

    Quote Originally Posted by HallsofIvy View Post
    Prove It was asking if you have the theorem that says "if lim_{x\to a} f(x)= F and \lim_{x\to a}g(x)= G then \lim_{x\to a} f(x)g(x)= FG. However, that does not appear to me to apply here since \lim_{x\to x_0} f(x) does not exist. Instead, it should be apparent that, in order that \lim_{x\to x_0) f(x)g(x) exist, the limit of g(x) must not only exist but must be a very specific number!
    why does it clear that in order the limit of f(x)*g(x) to be L, there's must to be a limit of g(x)? i didn't get that... how do you prove that?
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  6. #6
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    Re: help proving lim question

    The hypothesis is that \lim_{x\to x_0} f(x) is "infinity" which means that f(x) is very very large close to x_0. So what happens to f(x)g(x)? What must be true of g(x) so that f(x)g(x) does NOT get "very very large" close to x_0.
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  7. #7
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    Re: help proving lim question

    but can we multiple infinity by 0? i mean, if i got you right, you implied that in order that f(x)g(x) doesn't get very large (and to be L, a real number, as it's given), we need something to balance it -> lim g(x). and as i see it, it can only be done if lim g(x) exsits and equal 0, so then lim f(x)g(x) will equal 0 too.
    but if i'm not wrong, you can't multiple infinty by 0, can we?
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