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Homework Help: Limit as x->0

  1. Mar 31, 2008 #1
    1. The problem statement, all variables and given/known data
    Find the limit: lim x->0 (3/x^2 - 2/x^4)


    2. Relevant equations



    3. The attempt at a solution
    I've tried manipulating it in a lot of ways, but nomatter what I do I've still either got the limit of the denominator = 0 or one of the terms in either the num. or den. being divided by x (which is tending to zero)

    I've got as far as lim x->0 ( 9x^4 - 4 / 3x^6 + 2x^4 ) but this doesn't help as the den. = 0.

    I've graphed this function and had a look - the limit should equal -(infinity).

    Utterly confused.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 31, 2008 #2

    CompuChip

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    Welcome to PF caesius.

    Graphing the function was a good idea, at least now you know the answer :smile:

    You have
    [tex]\lim_{x\to0} \left( \frac{3}{x^2} - \frac{2}{x^4} \right)[/tex]
    Standard approach in these cases: write it as a single fraction. Though in principle you did this correctly, you could have done it much more easily by just multiplying the first term only by [itex]x^2 / x^2[/itex] and then adding the fractions. Of what you then get it is much easier to show that it tends to -infinity.
     
  4. Mar 31, 2008 #3
    Ok, I get (3x^2 - 2) / x^4 but I don't see how this helps.

    If x was tending to infinity I see how to solve it but for 0?
     
  5. Mar 31, 2008 #4

    CompuChip

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    Divide numerator and denominator by [itex]x^4[/itex].
     
  6. Mar 31, 2008 #5
    I've done that but I get 3x^-2 - 2x^-2 / 1

    Which to me looks like: (undefined) - (undefined) / 1

    How can that have a solution?
     
  7. Mar 31, 2008 #6

    HallsofIvy

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    Isn't "the limit does not exist" a solution?
     
  8. Mar 31, 2008 #7
    Ok I've just read how to format properly, for clarity I have:

    [tex]\frac{\frac{3}{x^2} - \frac{2}{x^2}}{1}[/tex]

    which when taking the limit as x tends to zero will result in indefined operations no?
     
  9. Mar 31, 2008 #8
    But I know the "limit" is [tex]-\infty[/tex], and I'm aware that this is not a limit by definition, surely I should be able to show this from the equation.
     
  10. Mar 31, 2008 #9

    HallsofIvy

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    Okay, here's what I would have done: combine the two fractions to get
    [tex]\frac{3}{x^2}- \frac{2}{x^4}= \frac{4x^2- 2}{x^4}[/tex]
    Since the numerator goes to -2 while the denominator goes to 0, the fraction itself goes to [itex]-\infty[/itex] (i.e. the limit does not exist). What more do you want?
     
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