# Two Limit exercises of functions of two variables.

• Thomasyar
In summary, a two limit exercise of a function of two variables involves finding the limit of a function with two independent variables as both variables approach a specific point simultaneously. This is done by evaluating the function at the point and observing the output as values get closer to the point. These exercises are important in mathematics as they allow us to understand the behavior of a function near a specific point. Common techniques used to solve two limit exercises include substitution, factoring, and algebraic manipulation, while more advanced techniques like L'Hôpital's rule and Taylor series may also be used. In real-life situations, two limit exercises of functions of two variables can be applied in fields such as physics and engineering to model real-world phenomena and make predictions about their
Thomasyar
(I) Find the limit (x,y)->(0,0) of F, then prove it by definition.

(II) Find the limit and prove it by definition of:

as (x,y) approach (C,0), C different from zero.
I have previously asked it on Quora, but it doesn't appear to have answers any time soon, and then I've encountered this forum.
--------------------

As every limit, the idea is to prove that it exist a positive Delta, which delimits the values of (x,y), for every Epsilon greater than zero. The common way to do it is to find a functional expression of Epsilon as a variable of the function Delta, where the domain of that function goes from zero to infinity, (0, +inf). Now, I have tried on both exercises multiple ways of findings relations between |(x,y)| and the corresponding function, all have failed.

In the second exercise the limit is obviously (0,0), due it doesn't have any discontinuity issue. But in the first one, I'm not totally sure if the limit exists. If it does, is (0,0).

The only thing that I explored deeply is delimiting the sine function as follows:
Abs(Sin(t)) <= Abs(t), and then t=x*y, since x <= |(x,y)| (same for y), so x*y <= |(x,y)|^2
(If 0 < C1 <= C3, and 0 < C2 <= C3, then C1*C2 <= C3*C2 <= C3^2; that’s what I used)
But that only lead me to have some sort of function of x and y in the denominator, and that is not useful.
(I managed to delimit |F(x,y)|<= 1/(sqrt(1+abs(y)))

Unless having some relation of the sort |F(x,y)|<= 1/J(x,y) is useful, I think gaining a relation of |(x,y)| in the denominator does not worth the try.

I would really appreciate it if someone gives me some help on proving those limits :)

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Thomasyar said:
(I) Find the limit (x,y)->(0,0) of F, then prove it by definition.
View attachment 219827
(II) Find the limit and prove it by definition of:
View attachment 219828
as (x,y) approach (C,0), C different from zero.
I have previously asked it on Quora, but it doesn't appear to have answers any time soon, and then I've encountered this forum.
--------------------
View attachment 219830

As every limit, the idea is to prove that it exist a positive Delta, which delimits the values of (x,y), for every Epsilon greater than zero.
The problems ask you to first find each limit, and then prove each limit using the definition of a limit (i.e., with ##\delta## and ##\epsilon##).
For the first limit I would change to polar form so that you can evaluate the limit as r approaches 0.
Thomasyar said:
The common way to do it is to find a functional expression of Epsilon as a variable of the function Delta, where the domain of that function goes from zero to infinity, (0, +inf). Now, I have tried on both exercises multiple ways of findings relations between |(x,y)| and the corresponding function, all have failed.

In the second exercise the limit is obviously (0,0), due it doesn't have any discontinuity issue.
No, the limit is not (0, 0). The limit for each function, if it exists, will be a single number.
Thomasyar said:
But in the first one, I'm not totally sure if the limit exists. If it does, is (0,0).

The only thing that I explored deeply is delimiting the sine function as follows:
Abs(Sin(t)) <= Abs(t), and then t=x*y, since x <= |(x,y)| (same for y), so x*y <= |(x,y)|^2
(If 0 < C1 <= C3, and 0 < C2 <= C3, then C1*C2 <= C3*C2 <= C3^2; that’s what I used)
But that only lead me to have some sort of function of x and y in the denominator, and that is not useful.
(I managed to delimit |F(x,y)|<= 1/(sqrt(1+abs(y)))

Unless having some relation of the sort |F(x,y)|<= 1/J(x,y) is useful,
I don't see that this is useful.
Thomasyar said:
I think gaining a relation of |(x,y)| in the denominator does not worth the try.

I would really appreciate it if someone gives me some help on proving those limits :)
BTW, the homework template that you deleted is required here. In future posts please don't delete this template.

tnich

## 1. What is a two limit exercise of a function of two variables?

A two limit exercise of a function of two variables is a mathematical problem that involves finding the limit of a function with two independent variables as both variables approach a specific point simultaneously.

## 2. How do you determine the limit of a function of two variables?

To determine the limit of a function of two variables, you must evaluate the function at the specific point and see if the output approaches a single value as both variables approach the point. This can be done by substituting values that are closer and closer to the point and observing the resulting output.

## 3. What is the importance of two limit exercises in mathematics?

Two limit exercises are important in mathematics because they allow us to understand the behavior of a function as it approaches a specific point. This can help us make predictions and draw conclusions about the function's behavior in that region.

## 4. What are some common techniques used to solve two limit exercises?

Some common techniques used to solve two limit exercises include substitution, factoring, and algebraic manipulation. In some cases, advanced techniques such as L'Hôpital's rule or Taylor series may also be used.

## 5. How can two limit exercises of functions of two variables be applied in real-life situations?

Two limit exercises of functions of two variables can be applied in real-life situations such as physics and engineering, where functions with two variables are often used to model real-world phenomena. These exercises can help us understand the behavior of these models and make predictions about their properties.

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