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Callisto

  • 2 years ago

If \(\lim_{x \rightarrow 0^{+}} f(x)=A\) and \(\lim_{x \rightarrow 0^{-}} f(x)=B\), find \(\lim_{x \rightarrow 0^{+}} f(x^3-x)\) How to start?

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  1. satellite73
    • 2 years ago
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    hmmm maybe find out whether \(x^3-x\) is approaching 0 from the right or the left as x approaches 0 from the right

  2. Callisto
    • 2 years ago
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    How....? FYI, I have the answer :\

  3. slaaibak
    • 2 years ago
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    What's the answer? lol

  4. slaaibak
    • 2 years ago
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    When in doubt, go with DNE, lol

  5. satellite73
    • 2 years ago
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    you can reason as follows: for small positive values of \(x\) we have \(x^3<x\) and so \(x^3-x<0\)

  6. satellite73
    • 2 years ago
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    this is my best guess at any rate i am trying to think up a counter example, one where you wouldn't know the limit, but off the top of my head i cannot, so perhaps what i wrote is correct

  7. satellite73
    • 2 years ago
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    in english, as \(x\to 0^+\) we have \(x^3-x\to 0^-\)

  8. satellite73
    • 2 years ago
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    so my guess is \(B\) although i have a 50% chance of being right even if my reasoning is faulty

  9. Callisto
    • 2 years ago
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    Nice *guess* :\

  10. satellite73
    • 2 years ago
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    thnx

  11. Callisto
    • 2 years ago
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    Assuming your way to do this question is correct. Similarly, for the question (in part b) \(\lim_{x \rightarrow 0^{-}} f(x^3-x)\) \[x^3-x>0\]So, as \(x \rightarrow 0^+\), \(x^3-x \rightarrow 0^{+}\). And it is A. Hmmm...

  12. Callisto
    • 2 years ago
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    *Assume

  13. UnkleRhaukus
    • 2 years ago
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    is the limit of the function of a sum , equal to the sum of the limits of the function ?

  14. Callisto
    • 2 years ago
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    As for part c, \[\lim_{x \rightarrow 0^{+}} f(x^2-x^4)\] \[x^2-x^4>0\] As \(x \rightarrow 0^{+}\), \(x^2-x^4\rightarrow 0^{+}\), so it is A. Seems this trick works, but I don't know why...

  15. HELP!!!!
    • 2 years ago
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    teach me how to do this

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