Derivation of wave equation using tension of a string

[Methias]

Homework Statement

I'm currently following the textbook Advanced Engineering Mathematics by Erwin Kreyszig.

I'm learning the derivation of the Wave equation using the method shown in the book, but when I reached the final part of the derivation, the working just confuses me.

(1/Δx)[ (u/dx)| x+Δx - (du/dx)| x ] = (p/T) ( second derivative of u wrt t )

at this part, it mentions that we are taking the lim ∆x→0, which will end up in the wave equation.

Homework Equations

(1/Δx)[ (u/dx)| x+Δx - (du/dx)| x ]

The Attempt at a Solution

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I've tried working out the limit, but I guess there's something I don't know since as Δx approaches zero, the derivative terms will become the same and cancel out each other?

 

[DrClaude]

$$
\lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = \frac{df}{dx}
$$