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Math Help - Squeeze Theorem

  1. #1
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    Squeeze Theorem

    During class today we started covering some stuff that had me totally lost. We were covering The squeeze theorem and I was doing really good up until
    lim xsin 1/x
    I haven't the faintest idea what he did or how to repeat it. Can anyone give me a thorough explanation of this? The guy speaks terrible english and after his very brief run through of it simply told us to do
    lim x^2 sin 1/x
    lim x^2 cos 1/x
    at home and know how to do them for tomorrows quiz.
    So any advise on where to begin with those would be awesome. I think i'm mostly lost bc I don't remeber the trig functions i need. (like tanx=sinx/cosx). But I may be way off. Help
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  2. #2
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    note from the graph that -|x| \le x \sin\left(\frac{1}{x}\right) \le |x|

    so ... what can you say about the limit of the function in question as x \to 0 ?
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  3. #3
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    I don't understand where you come up with the x and -x at all. I got to the graph part and don't know what i'm looking for from there. And that doesn't help me understand what to do with the other 2 problems.
    x^2sin1/x
    x^2cos1/x
    which are my main concern.
    I also understand what the answer of xsin1/x is. i just need to know how to get there.
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  4. #4
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    Are you taking the limit as x approaches 0?

    First, we know that |\sin \frac{1}{x}| \le 1 thus it should be clear that 0 \le |x \sin \frac{1}{x}| \le |x| \Leftrightarrow -|x| \le x \sin \frac{1}{x} \le |x|

    Now, \lim_{x\to 0} |x| = 0. From here apply the squeeze theorem.
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