How to Find Points of Discontinuity for a 2-Variable Limit in Calculus 3?

Thread starter lukasz08
Start date Nov 11, 2008
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Calculus Calculus 3 Limit Variable

In summary, a limit of a function of two variables is a value that the function approaches as the two variables approach a specific point on the function's domain. It can be used to determine continuity, differentiability, and other properties of the function. To evaluate a limit, you can use techniques such as substitution and algebraic manipulation. The main difference between a limit of a function of two variables and a limit of a function of one variable is that the former involves two independent variables. A limit of a function of two variables can exist even if the function is not defined at that point, and it is commonly used in real-life applications to model and predict the behavior of complex systems.

Nov 11, 2008

lukasz08

find the points of discontinuouty

limit as x,y ---> (0,0) of function f(x,y) = 1-((cos(x^2+y^2)/(x^2+y^2))

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Nov 12, 2008

Pere Callahan

What have you tried so far? And are you sure that you don't have a typo in your function? As it is, it diverges as (x,y)->(0,0).

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

A limit of a function of two variables is a value that a function approaches as the two variables approach a specific point on the function's domain. It represents the behavior of the function near that point and can be used to determine continuity, differentiability, and other properties of the function.

2. How do you evaluate a limit of a function of two variables?

To evaluate a limit of a function of two variables, you can use the same techniques as for a function of one variable. You can substitute values for the two variables and observe the resulting output values, or you can use algebraic techniques such as factoring, rationalizing, and simplifying to find the limit.

3. What is the difference between a limit of a function of two variables and a limit of a function of one variable?

The main difference between a limit of a function of two variables and a limit of a function of one variable is that the former involves two independent variables, while the latter only has one. This means that the behavior of the function near a specific point depends on the values of both variables, rather than just one.

4. Can a limit of a function of two variables exist even if the function is not defined at that point?

Yes, a limit of a function of two variables can exist even if the function is not defined at that point. This is because the limit only considers the behavior of the function near the point, not the actual value at the point itself. However, if the limit does not exist, it also means that the function is not defined at that point.

5. How is the concept of a limit of a function of two variables used in real-life applications?

The concept of a limit of a function of two variables is used in many real-life applications, particularly in fields such as physics, engineering, and economics. It can be used to model and predict the behavior of complex systems with multiple variables, such as fluid flow, heat transfer, and optimization problems.