Question regarding the derivation of the wave equation

[Chump]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[andrewkirk]

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.