Finding an infinitesimal limit function

[greswd]

TL;DR Summary

A function which is the infinitesimal limit of another function

I have the function: $\sqrt{\left(\frac{x}{h}+1\right)^{2}+\left(\frac{y}{h}\right)^{2}}-\sqrt{\left(\frac{x}{h}\right)^{2}+\left(\frac{y}{h}\right)^{2}}$

I would like to find an analytical solution, the equivalent function, in the limit of h approaching zero.


Additional info which might be useful, there's the function: $\frac{1}{k}\left(\sqrt{\frac{\left(\left(1-k^{2}\right)\cdot y\right)^{2}}{x^{2}+\left(1-k^{2}\right)\cdot y^{2}}+\left(\frac{x}{\sqrt{x^{2}+\left(1-k^{2}\right)\cdot y^{2}}}+k\right)^{2}}-1\right)$

I believe that it also approaches the same solution function in the limit of k approaching zero.

 

[PeroK]

You can calculate the limit by taking out a factor of $\frac 1 t$, then using the usual technique of multiplying by the conjugate $\sqrt X + \sqrt Y$ etc.

 

[mfb]

$$\sqrt{a}-\sqrt{b} = \frac{a-b}{\sqrt{a}+\sqrt{b}}$$
The right hand side will make it easier to evaluate the limit.

 

[PeroK]

$$\sqrt{a}-\sqrt{b} = \frac{a-b}{\sqrt{a}+\sqrt{b}}$$
The right hand side will make it easier to evaluate the limit.

That's what I meant!

 

[greswd]

awesome, I was able to calculate it, $\frac{x}{\sqrt{x^{2}+y^{2}}}$

 

[greswd]

how do you think I can reduce the 2nd equation?

 

[mfb]

The same approach should work there. $1=\sqrt{1}$ and you have a difference of square roots again.