Limit as a variable of two variables

In mathematics, limits play a crucial role in understanding the behavior of functions and their values as they approach certain points or infinity. While limits are often associated with functions of a single variable, they can also be extended to functions of two variables. This opens up a whole new realm of possibilities and challenges in understanding the limits of such functions.

Let’s consider a function f(x, y) of two variables x and y. In this case, a limit is defined as the value that the function approaches as it gets arbitrarily close to a certain point (x₀, y₀) in its domain. To determine the limit, we explore the behavior of the function for different paths or approaches to the point (x₀, y₀).

The concept of limit as a variable of two variables becomes particularly interesting when dealing with more complex functions. Consider a function f(x, y) = (x² – y²) / (x + y). If we try to evaluate it directly at the point (0, 0), we encounter a problem as it results in an indeterminate form, since division by zero is undefined. However, this indeterminacy can be resolved by examining the limits as x and y approach zero through different paths.

Let’s analyze the limit of f(x, y) as (x, y) approaches (0, 0) along the x-axis. In this case, y is fixed at zero, and as x gets closer to zero, we observe that the numerator, (x² – 0²), approaches zero. Simultaneously, the denominator, (x + 0), also approaches zero. Now, if we divide zero by zero, it is still indeterminate. However, by factoring the numerator as (x + y)(x – y) and simplifying, we obtain the limit of f(x, y) as x approaches zero along the x-axis as y. This limit evaluates to zero, indicating that f(x, y) approaches zero in this specific direction.

Similarly, let’s explore the limit of f(x, y) as (x, y) approaches (0, 0) along the y-axis. In this case, x is fixed at zero, and as y approaches zero, both the numerator and denominator become zero. By factoring again and simplifying, we can find the limit as y approaches zero along the y-axis. This limit also evaluates to zero.

Interestingly, if we consider a different path, such as along the line y = x, we get a different result. In this case, as (x, y) approaches (0, 0) along this line, the numerator simplifies to (x² – x²) = 0, while the denominator becomes (x + x) = 2x. Therefore, the limit as (x, y) approaches (0, 0) along the line y = x is zero divided by 2x, which is zero.

As we delve into these examples, it becomes clear that the limit as a variable of two variables is not necessarily unique. The limit depends on the path of approach to the specified point (x₀, y₀). If the limit is the same for all possible paths, then the function is said to have a limit at that point. In our example, f(x, y) has a limit of zero as (x, y) approaches (0, 0) along different paths, indicating that it possesses a limit at that point.

In conclusion, extending the concept of limits to functions of two variables allows us to study the behavior of these functions as they approach specific points. The limit as a variable of two variables can have various values depending on the path of approach. Understanding these limits provides valuable insights into the behavior and properties of functions in higher dimensions.