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Math Help - Limits

  1. #1
    Newbie
    Joined
    Sep 2008
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    Limits

    hey I got this question:


    Question
    lim x sin1/x=0
    x->0
    ____________

    answer
    -1<=sin(1/x)<=1

    -x<=xsin(1/x)<=x

    lim -x = 0
    x->0

    lim x=0
    x->0

    -->lim x sin(1/x)=0 //MY answer
    x->0

    __________________

    I just want to know if my working and answer is correct!tnx
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Santa Cruz, CA
    Posts
    2,844
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    Quote Originally Posted by geeko View Post
    hey I got this question:


    Question
    lim x sin1/x=0
    x->0
    ____________

    answer
    -1<=sin(1/x)<=1

    -x<=xsin(1/x)<=x

    lim -x = 0
    x->0

    lim x=0
    x->0

    -->lim x sin(1/x)=0 //MY answer
    x->0

    __________________

    I just want to know if my working and answer is correct!tnx
    The answer is correct, and the working seems to be correct too [since I'm not an expert on applying the Squeeze Theorem]

    Here's another approach:

    \lim_{x\to0}x\sin\left(\frac{1}{x}\right)

    Let z=\frac{1}{x}\implies x=\frac{1}{z}

    Now, as x\to 0,~z\to\infty

    Thus, \lim_{x\to 0}x\sin\left(\frac{1}{x}\right)=\lim_{z\to\infty}\  frac{\sin z}{z} which converges to zero.

    --Chris
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  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
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    Quote Originally Posted by Chris L T521 View Post
    The answer is correct, and the working seems to be correct too [since I'm not an expert on applying the Squeeze Theorem]

    Here's another approach:

    \lim_{x\to0}x\sin\left(\frac{1}{x}\right)

    Let z=\frac{1}{x}\implies x=\frac{1}{z}

    Now, as x\to 0,~z\to\infty

    Thus, \lim_{x\to 0}x\sin\left(\frac{1}{x}\right)=\lim_{z\to\infty}\  frac{\sin z}{z} which converges to zero.

    --Chris

    how about this question

    lim f(x) where |f(x)-1| <=x2(p.s x square) , x not equal to 0
    x->0

    multiply by x ..... -x <= x sin(1/x) <= x

    lim-x=0
    x->0

    lim x=0
    x->0

    therefore lim x sin(1/x)=0
    x->0

    Is it correct too?


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