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Math Help - Limits by definition.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Limits by definition.

    Prove by definition (epsilon-delta) the next limits:

    1. lim(x-->inf) {(2x^2-11x-11)/(4x^2+11x-1)} = 1/2

    2. lim(x-->1) {(x^3-3x)/(x^2-2x+1)} = -inf



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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Prove by definition (epsilon-delta) the next limits:

    1. lim(x-->inf) {(2x^2-11x-11)/(4x^2+11x-1)} = 1/2

    2. lim(x-->1) {(x^3-3x)/(x^2-2x+1)} = -inf



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    For the first one note that, \frac{2x^2-11x-11}{4x^2+11-1x}=\frac{2-\frac{11}{x}-\frac{11}{x^2}}{4+\frac{11}{x}-\frac{1}{x^2}}. Now, \left|2-\frac{11}{x^2}-\frac{11}{x}-2\right|=\left|\frac{11}{x^2}+\frac{11}{x}\right|\  leqslant\frac{22}{x}. So, let \varepsilon>0 be given. Choosing x>\frac{22}{\varepsilon} ensures that \left|2-\frac{11}{x^2}-\frac{11}{x}\right|\leqslant\frac{22}{x}<\frac{22}  {\frac{22}{\varepsilon}}=\varepsilon. We may conclude that 2-\frac{11}{x^2}-\frac{11}{x}\to2. Using similar techniques we may conclude that 4+\frac{11}{x}-\frac{1}{x^2}\to4. We may therefore conclude that \lim_{x\to\infty}\frac{2x^2-11x-11}{4x^2+11x-x}=\lim_{x\to\infty}\frac{2-\frac{11}{x}-\frac{11}{x^2}}{4+\frac{11}{x}-\frac{1}{x^2}}=\frac{2}{4}=\frac{1}{2}

    For the second one, note that \frac{x^3-x}{x^2-2x+1}=2+x+\frac{2}{x-1} which for a sufficiently small neighborhood around 1 is strictly less than 3+\frac{2}{x-1}. So, let T<0 be given, then choosing 1>x>\frac{T-1}{T-3} ensures that 3+\frac{2}{x-1}<3+\frac{2}{\frac{T-1}{T-3}}=T. The conclusion follows.

    P.S. Technically, the second should be \lim_{x\to 1^{-}}
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    \left|2-\frac{11}{x^2}-\frac{11}{x}-2\right|=\left|\frac{11}{x^2}+\frac{11}{x}\right|\ leqslant\frac{22}{x} Wy is that? |2-....-2| ???
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    \left|2-\frac{11}{x^2}-\frac{11}{x}-2\right|=\left|\frac{11}{x^2}+\frac{11}{x}\right|\ leqslant\frac{22}{x} Wy is that? |2-....-2| ???
    What??????????????????????????????????????????
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    Sorry!
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Sorry!
    Haha. Don't be sorry! I knew you just didn't look close enough!
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