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Math Help - evaluating a limit

  1. #1
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    evaluating a limit

    I need help solving this one guys:
    \lim_{x\to -\infty}\frac{2x}{3\sqrt{x^2+1}}
    How can I show that the limit exists and prove it?
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  2. #2
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    Quote Originally Posted by binkypoo View Post
    I need help solving this one guys:
    \lim_{x\to -\infty}\frac{2x}{3\sqrt{x^2+1}}
    How can I show that the limit exists and prove it?
    Note that \frac{2x}{3\sqrt{x^2+1}} = \frac{2x}{3 |x| \sqrt{1 + \frac{1}{x^2}}} and that \frac{x}{|x|} = -1 for this limit (why?).
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  3. #3
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    ok so now my strategy is to use the squeeze theorem to show that the function \frac{x}{|x|\sqrt{1+1/(x^2)}} goes to -1 as x goes to -\infty.
    As you suggested, x/|x| goes to -1. My problem now is finding something smaller than \frac{x}{|x|\sqrt{1+1/(x^2)}} which also goes to -1 as x goes to -\infty

    Is this even the right strategy to take? Im a little lost.
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  4. #4
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    Quote Originally Posted by binkypoo View Post
    ok so now my strategy is to use the squeeze theorem to show that the function \frac{x}{|x|\sqrt{1+1/(x^2)}} goes to -1 as x goes to -\infty.
    As you suggested, x/|x| goes to -1. My problem now is finding something smaller than \frac{x}{|x|\sqrt{1+1/(x^2)}} which also goes to -1 as x goes to -\infty

    Is this even the right strategy to take? Im a little lost.
    I'm sure it's sufficient to note that \frac{1}{x^2} \rightarrow 0 in the limit of x \rightarrow - \infty and use a couple of basic limit theorems (it appears that you have more or less done that in saying that
    \frac{x}{|x|\sqrt{1+1/(x^2)}} goes to -1 as x goes to -\infty.
    .
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  5. #5
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    well Im trying to use the squeeze theorem, but I cant find a function smaller than x/(|x|sqrt(1+1/x^2))) which goes to -1 as x goes to -inf.
    So I havent actually shown that x/(|x|sqrt(1+1/x^2))) goes to -1 yet!
    I dont quite see how showing 1/x^2 going to 0 would be an effective approach, could you elaborate? Thanks for thehelp
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  6. #6
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    Quote Originally Posted by binkypoo View Post
    well Im trying to use the squeeze theorem, but I cant find a function smaller than x/(|x|sqrt(1+1/x^2))) which goes to -1 as x goes to -inf.
    So I havent actually shown that x/(|x|sqrt(1+1/x^2))) goes to -1 yet!

    I dont quite see how showing 1/x^2 going to 0 would be an effective approach, could you elaborate? Mr F says: Well, how did you establish the limit in the first line of post #3 without using this ....?

    Thanks for thehelp
    All you have to do is prove that \lim_{x \rightarrow - \infty} \frac{1}{\sqrt{1 + \frac{1}{x^2}}} = 1 and this clearly follows from the fact that \lim_{x \rightarrow - \infty} \frac{1}{x^2} = 0.
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