Heat Wave Equations: Explaining Delta t & x Approach to Zero

[Kajan thana]

TL;DR

Hi Guys, I am trying to understand the derivation of the diffusion/heatwave equation, but I am stuck on how the person managed to get $k^2$. I have attached the slide to this thread.

When the delta t and x approached zero, assumably it results in the whole fraction to be zero. The slide shows it will be $k^2$. Can someone explain this to me, please?

P.S. I have tried to search this up, I could not find anything related to the confusion.

 

[mfb]

Are you sure this isn't simply the definition of $\kappa$? The limit is not negative, so you can define $\kappa$ to be the square root of it. It's a kappa, not k.

 

[Kajan thana]

Are you sure this isn't simply the definition of $\kappa$? The limit is not negative, so you can define $\kappa$ to be the square root of it. It's a kappa, not k.

First of all , thank you for the instant reply! But I am still confused.. is that the mathematical definition for the $\kappa$?

 

[mfb]

I think so. The limit is some value, you define $\kappa$ to be the square root of that limit, i.e. the limit is $\kappa^2$.