A multivariable limit is a mathematical concept that describes the behavior of a function as it approaches a certain point in a multi-dimensional space. It is typically denoted as lim(x,y)→(a,b)f(x,y), where f(x,y) is the function and (a,b) is the point in question.
To evaluate a multivariable limit, you must first determine if the function is continuous at the point in question. If it is, then you can simply plug in the values of (a,b) into the function to find the limit. If the function is not continuous, then you will need to use more advanced techniques such as L'Hospital's rule or algebraic manipulation to evaluate the limit.
The main difference between a multivariable limit and a single variable limit is the number of dimensions involved. A single variable limit only considers the behavior of a function along a single axis, whereas a multivariable limit takes into account the behavior of a function along multiple axes in a multi-dimensional space.
Yes, it is possible for a multivariable limit to exist but not be equal to the value of the function at that point. This typically occurs when the function is not continuous at the point in question, and the limit represents the behavior of the function as it approaches the point rather than the value at the point itself.
Multivariable limit evaluation is commonly used in fields such as physics, engineering, and economics to model and analyze complex systems that involve multiple variables. It is also used in optimization problems to find the maximum or minimum value of a function in a multi-dimensional space.