# Finding an infinitesimal limit function

• I
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.

Last edited:

Homework Helper
Gold Member
2022 Award
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.

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

• PeroK
Homework Helper
Gold Member
2022 Award
$$\sqrt{a}-\sqrt{b} = \frac{a-b}{\sqrt{a}+\sqrt{b}}$$
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}}}##

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

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