Limit for a two variables function Question

In summary, the conversation is about a person searching for help in solving an exercise involving calculating the existence of a limit in (0,0) for two functions. They have attempted the first one and have reached a solution, but are unsure if they have done it correctly. They mention using a definition to calculate the limit and ask for help with the second exercise. The person responding suggests converting to polar coordinates to find the limit.
  • #1
Freydulf
2
0
Hi!

I have some doubts bound up with an exercise of limits. It demands to calculate the existence of the limit in (0,0) of these two functions.

http://www.rinconmatematico.com/latexrender/pictures/bd1b868f7e07af600ea0c82b34a38137.png [Broken]

http://www.rinconmatematico.com/latexrender/pictures/5a3966109b20a098800453a99467592c.png [Broken]

My solution for the first one til now is:

http://www.rinconmatematico.com/latexrender/pictures/667c03805a5b4b203406f69cf574c56e.png [Broken]

Well, after the substitution we have http://www.rinconmatematico.com/latexrender/pictures/04dd2173792059a5b0e5447b852baf28.png [Broken]

So we can use http://www.rinconmatematico.com/latexrender/pictures/722e499f2e8ad1b9b66060f151689378.png [Broken]

http://www.rinconmatematico.com/latexrender/pictures/2bde7e5f3e31f640f734b26dcf229dc4.png [Broken]

Using http://www.rinconmatematico.com/latexrender/pictures/03d4bbee51cd042a5e43f4b8eed0e81e.png [Broken]

http://www.rinconmatematico.com/latexrender/pictures/a4abe4b98d7e1c4782faca7c9db897f9.png [Broken]

And http://www.rinconmatematico.com/latexrender/pictures/597c072a27536c4904c39673dc916ace.png [Broken]

http://www.rinconmatematico.com/latexrender/pictures/b04c33391868c0ff16d9fa2933c6466b.png [Broken]

Well, I don't know if I've done correctly all the process, but if I did, I guess that in the next step I have to calculate the limit by the definition to conclude the exercise, and that's one of the things that I don't know how to do.
The second exercise is similar to the first one but I don't know how to do it either. Maybe the minus changes the result and there's not limit and any necessity of employing the definition.
Anyhow, can someone lend me a hand? :)

Thanks!
 
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  • #2
For problems like this, after you have tried taking the limit along different paths and seen that you don't get different answer (which would have proved the limit does not exist), I recommend converting to polar coordinates. That way "closeness" to (0,0) is measured only by r, not the angle [/itex]\theta[/itex]. If the limit, as r goes to 0, is a constant, not depending on [itex]\theta[/itex] that is the limit of the function. If the limit as r goes to 0 depends on [itex]\theta[/itex], the limit does not exist.
 

1. What is a limit for a two variables function?

A limit for a two variables function is a mathematical concept that describes the behavior of a function as the independent variables approach a certain value. It helps us understand the behavior and properties of functions in more complex situations.

2. How is a limit for a two variables function calculated?

The limit for a two variables function is calculated by considering the values of the function as the independent variables approach a certain value. This can be done by evaluating the function at different points and observing the trend of the values as the variables get closer to the desired value. Calculus techniques can also be used to find the limit analytically.

3. What is the importance of finding the limit for a two variables function?

Finding the limit for a two variables function is important because it helps us understand the behavior of a function near a specific point. It allows us to make predictions and approximations about the values of the function, and also helps us identify any discontinuities or asymptotes in the function.

4. Can a limit for a two variables function have more than one value?

No, a limit for a two variables function can only have one value. This value represents the behavior of the function as the variables approach a certain value and does not depend on the path taken to reach that value. If the limit has more than one value, then the function is said to be discontinuous at that point.

5. How is a limit for a two variables function used in real-world applications?

Limits for two variables functions are used in many real-world applications, such as physics, engineering, and economics. They can help model and predict the behavior of complex systems, such as the motion of objects, the flow of fluids, and the growth of populations. They are also used in optimization problems to find the maximum or minimum value of a function.

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