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:
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