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Math Help - Using the Squeeze Theorem on Cosine Function

  1. #1
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    Using the Squeeze Theorem on Cosine Function

    I'm trying to understand the proof of this function:

    <br />
\displaystyle \lim_{x\to 0 } x^2cos 20 \pi x =0}<br />

    Here is the part that I understand:

    Let f(x)=-x^2, g(x)=x^2cos 20 \pi x, h(x)=x^2
    Then, -1 \leq cos 20 \pi x \leq 1

    But then, they go on to state this, which I can't figure out:

    -x^2 \leq x^2cos 20 \pi x \leq x^2

    What I don't understand is how the x-squared terms can always be either less than -1 or greater than 1. I understand that the middle term oscillates between -1 and 1, but how do we know what x-squared is?

    Can anybody help me understand this?

    Thanks.
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  2. #2
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    Quote Originally Posted by joatmon View Post
    I'm trying to understand the proof of this function:

    <br />
\displaystyle \lim_{x\to 0 } x^2cos 20 \pi x =0}<br />

    Here is the part that I understand:

    Let f(x)=-x^2, g(x)=x^2cos 20 \pi x, h(x)=x^2
    Then, -1 \leq cos 20 \pi x \leq 1

    But then, they go on to state this, which I can't figure out:

    -x^2 \leq x^2cos 20 \pi x \leq x^2

    What I don't understand is how the x-squared terms can always be either less than -1 or greater than 1. I understand that the middle term oscillates between -1 and 1, but how do we know what x-squared is?

    Can anybody help me understand this?

    Thanks.
    \displaystyle\lim_{x\to 0} -x^2=\lim_{x\to 0}x^2=0

    By the squeeze theorem, the overall limit must be 0.

    Did that help?
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  3. #3
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    I did not understand what is your problem.
    However, the second inequality formed by multiplying the first one by \displaystyle x^2.
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  4. #4
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    Here is the graph of the squeeze theorem in action.

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  5. #5
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    Quote Originally Posted by joatmon View Post
    I'm trying to understand the proof of this function:

    <br />
\displaystyle \lim_{x\to 0 } x^2cos 20 \pi x =0}<br />

    Here is the part that I understand:

    Let f(x)=-x^2, g(x)=x^2cos 20 \pi x, h(x)=x^2
    Then, -1 \leq cos 20 \pi x \leq 1

    But then, they go on to state this, which I can't figure out:

    -x^2 \leq x^2cos 20 \pi x \leq x^2

    What I don't understand is how the x-squared terms can always be either less than -1 or greater than 1. I understand that the middle term oscillates between -1 and 1, but how do we know what x-squared is?

    Can anybody help me understand this?

    Thanks.
    There is nothing there that says that "the x-squared terms can always be either less than -1 or greater than 1." Where did you get that idea? -x^2\le x^2cos(20\pi x)\le x^2 says nothing about how large x^2 is.

    The point is to take the limit as x goes to 0. As x goes to 0, so does x^2.
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  6. #6
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    Thanks for the help!
    Last edited by joatmon; January 22nd 2011 at 10:47 AM.
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