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Yet another limit problem!

  1. May 2, 2005 #1
    I was told to differentiate by definition!!! Function
    f = 1/(sqrt(1+x^2));

    which gives me the expression
    Lim(h->0) {(1/(sqrt(1+(x+h)^2)) - 1/(sqrt(1+x^2)))/h}
    problem is...I can’t seem to get rid of the h at the bottom...I’ve tried all teh math packages I have including maple and all of them just seem to complicate things! what should be the simplified version of that function so I don’t get h at the bottom resulting in undefined numbers. or is there a different way to look at this definition problem?

    Thanks heaps
    K. Civilian
     
  2. jcsd
  3. May 2, 2005 #2

    matt grime

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    Why would you want to get rid of it? you need to let it tend to zero....


    This ought to help:


    Edit:


    sqrt(a)-sqrt(b)= (a-b)/(sqrt(a)+sqrt(b))
     
    Last edited: May 2, 2005
  4. May 2, 2005 #3

    uart

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    I suggest you start like this :

    ( 1 + (x+h)^2 )^(-1/2)

    = ( 1 + x^2 + (2x+h)h )^(-1/2)

    = (1 + x^2)^(-1/2) ( 1 + (2x+h)/(1+x^2) h )^(-1/2)

    Then do a binomial expansion of the "(1 + (2x+h)/(1+x^2) h)^(-1/2)" factor and it's pretty straight forward from there.
     
  5. May 2, 2005 #4

    HallsofIvy

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    Because if it is in the denominator as it goes to 0 it will cause a lot of trouble!


    Provided a-b= 1?
     
  6. May 2, 2005 #5

    matt grime

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    Noted my mistake.

    But the point is that if you use that corrected identity, then you get something where the h on bottom cancels off, and you just have an h on top, let it go to zero and the correct derivative drops out.

    Since it's easier to typeset, here's the idea of 1/x

    1/(x+h) - 1/x = -h/(x+h)(x)

    divide through by h and now let he tend to zero to see that the derivative is -1/x^2

    that works here too, but it's a bugger to typeset.
     
    Last edited: May 2, 2005
  7. May 2, 2005 #6

    HallsofIvy

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