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Question regarding the derivation of the wave equation

  1. Oct 3, 2015 #1
    1. The problem statement, all variables and given/known data
    There's a derivation here that I'm looking at, and I've hit a snag. At (1) about 15 lines down the page, the author divides by Δx and takes the limit as Δx goes to 0. I understand what he did on the right side of the equation, but on the left side of the equation, by what means is he able to transform √(Δx2 + Δu2) into √(1 + (du/dx)2)? I'm kind of lost there.

    http://www.math.ubc.ca/~feldman/m256/wave.pdf

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Oct 3, 2015 #2

    andrewkirk

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    It uses the following property of limits:

    The limit of the n-th root of function f(x) exists and is equal to the n-th root of the limit of f(x), if the latter exists.

    Here n is 2.

    The property is often taught in teaching limits, and is easy to prove using epsilon-delta arguments.
     
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