# Finding an infinitesimal limit function

• I
• greswd
In summary, the conversation discusses finding an analytical solution for a given function in the limit of a certain variable approaching zero. The function is expressed in terms of square roots and the limit can be calculated by taking out a factor and using the technique of multiplying by the conjugate. The same approach can be used for a related function in the conversation.
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:
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.

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

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

That's what I meant!

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

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

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

## 1. What is an infinitesimal limit function?

An infinitesimal limit function is a mathematical concept used in calculus to describe the behavior of a function as its input approaches a certain value. It represents the value that the function approaches but never quite reaches, known as the limit.

## 2. How do you find an infinitesimal limit function?

To find an infinitesimal limit function, you must first evaluate the function at various points close to the limit value. Then, you can use algebraic techniques or graphical methods to determine the behavior of the function as the input approaches the limit value.

## 3. What is the importance of an infinitesimal limit function?

Infinitesimal limit functions are important in calculus because they allow us to analyze the behavior of a function at a specific point, even if the function is undefined at that point. They also help us understand the relationship between different functions and their limits.

## 4. What are some common techniques for finding an infinitesimal limit function?

Some common techniques for finding an infinitesimal limit function include using algebraic manipulation, L'Hôpital's rule, and the squeeze theorem. Graphical methods, such as using a graphing calculator or drawing a graph, can also be helpful.

## 5. Are there any limitations to finding an infinitesimal limit function?

Yes, there are limitations to finding an infinitesimal limit function. In some cases, the limit may not exist or may be infinite. Additionally, some functions may be too complex to evaluate using traditional methods, requiring more advanced techniques such as Taylor series or numerical methods.

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