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Math Help - Sandwich Theorem

  1. #1
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    Sandwich Theorem

    My teacher just touched on this today. I am trying to figure out some problems. I will show you one, and show how I did it. Could you tell me if I am doing this right? I could be totally off. Thanks in advance.


    Find the limit and confirm your answer using the sandwich theorem.


    lim as x approaches infinity of (1-cosx)/(x^2)

    -1 <= cos x <= 1

    2 <= 1-cosx <= 0

    (2/x^2) <= (1-cosx)/(x^2) <= 0/(x^2)

    0 <= (1-cosx)/(x^2) <= 0

    So, the answer is 0.


    ???
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by Morgan82 View Post
    My teacher just touched on this today. I am trying to figure out some problems. I will show you one, and show how I did it. Could you tell me if I am doing this right? I could be totally off. Thanks in advance.


    Find the limit and confirm your answer using the sandwich theorem.


    lim as x approaches infinity of (1-cosx)/(x^2)

    -1 <= cos x <= 1

    2 <= 1-cosx <= 0

    (2/x^2) <= (1-cosx)/(x^2) <= 0/(x^2)

    0 <= (1-cosx)/(x^2) <= 0

    So, the answer is 0.


    ???
    -1 \leq \cos x \leq 1
    1 \geq -\cos x \geq -1

    add 1 to all three sides:

    2 \geq 1 - \cos x \geq 0

    Your reasoning looks okay -- just make sure you justify the way your teacher expects you to.
    Good luck!!
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  3. #3
    Junior Member
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    yup, do take note of your careless mistake. how can f(x) be bigger then 2 and smaller then 0? good luck! =)

    1 <= cos x <= 1

    2 <= 1-cosx <= 0
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