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Homework Help: Finding the limit

  1. Feb 22, 2009 #1
    1. The problem statement, all variables and given/known data

    find the limit if it exist :
    lim (1/[x+Δχ] - 1/x) / Δχ

    3. The attempt at a solution
    lim (1/[x+Δχ] - 1/x) / Δχ => lim -Δχ /(x+ Δx) / Δχ
    then what ? do i cancel the ΔX in the numerator with that in the denominator ? or whats the next step to solve this problem and did I do any mistake?
  2. jcsd
  3. Feb 22, 2009 #2


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    That limit is undefined as x->0. Are you sure you don't mean delta X ->0? And, if so, sure, you want to cancel the thing that's going to 0. And don't write things like a/b/c without parentheses. It's not clear whether you mean (a/b)/c or a/(b/c). They are different.
  4. Feb 22, 2009 #3
    oh yes, I'm sorry , I miss typed it its supposed to be delta x approaching 0 from the left hand side of the graph.
  5. Feb 22, 2009 #4


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    Ok. Let's write h instead of delta X, ok. You want limit h->0 (1/(x+h)-1/x)/h. I'm not really happy with -h/(x+h)/h for reasons beyond the parentheses.
  6. Feb 22, 2009 #5
    You have 1/(x+dx) - 1/x

    Cross multiply them, then see what you're left with.

    edit: Oh you already did that, the dx cancels and as dx->0 it tends to -1/x^2 no?
  7. Feb 22, 2009 #6
    it matches my answer but i wanted to make sure that i did the proper steps
  8. Feb 22, 2009 #7


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    I think I understand what you meant to say, but cross multiplication applies when you have an equation with two rational expressions, such as
    a/b = c/d

    "Cross multiplication" results in ad = bc, and is equivalent to multiplying both sides of the equation by bd.
  9. Feb 22, 2009 #8
    I mean

    x/x(x+dx) - (x+dx)/x(x+dx) to get -dx/(x^2+dx)
  10. Feb 23, 2009 #9


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    He meant, "get a common denominator and subtract the two fractions."
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