Saturday, November 9, 2024

Limit of a function of three or more variables

 Function of three or more variables

Let's consider two functions respectively of three and four variables that have applications in the real world. Let' f(x,y,z) be a function that measures the temperature at a location (x,y,z) and f(x,y,z,t) a function that measures the air pressure at a time t. How can we take the limit of a function in R³? What does it mean for a function of three or more variables to be continuous?

The answers to these questions start by extending the notion of a δ disk in a function of two variables to functions of three or more variables. The notions of limits and continuity for these functions are similar to those of a function of two variables.

Definition of a δ ball

The notion of δ intervenes in the definition of a function in any dimension. It is a neighborhood located in the domain of definition of a function. In a function of one variable, it is represented by a line. This line represents al the points x of which the distance from a central point is less than ẟ. In a function of two variables, it is represented by a disk. This disk represents all the points located within the distance of the radius of a circle centered a point (x₀, y₀) than In a function of two or more variables it is represented by a ball which represents all the points located with the distance of the radius of the ball. The ball is also called sphere. Here is the definition:

 










Limit of a function of three or more variables

The limit of a function implies two neighborhoods: one in the domain of the function and the other in the range. To show that the limit of a function at a point located in the domain of the function exists, we need to find a neighborhood located around that point such that for any point located in that neighborhood the value of the function at that point is close to a fixed value  called the limit of the function. This consideration is valuable for a function in any dimension.

In two dimensions the definition states as follows:




In three dimensions we have:









In 4 dimensions we have:























Example 







Solution





Practice

Find the limit of the following function:
















 


No comments: