%2F(x%5E2%2By%5E2)-continuous.jpeg "(x^2-y^2)/(x^2+y^2) Continuous")
Exploring the Continuity of (x^2 - y^2) / (x^2 + y^2)
This article delves into the continuity of the function f(x, y) = (x^2 - y^2) / (x^2 + y^2). We will analyze its behavior and identify where it is continuous and where it experiences discontinuities.
Understanding Continuity
In calculus, a function is considered continuous at a point if its graph can be drawn without lifting the pen. More formally, a function is continuous at a point (a, b) if:
- f(a, b) is defined: The function has a value at the point.
- lim (x, y) → (a, b) f(x, y) exists: The limit of the function as (x, y) approaches (a, b) exists.
- lim (x, y) → (a, b) f(x, y) = f(a, b): The limit and the function value are equal.
Analyzing the Function
Let's examine the function (x^2 - y^2) / (x^2 + y^2):
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Domain: The function is defined for all real values of x and y except when the denominator becomes zero. This happens when x^2 + y^2 = 0, which only occurs at the point (0, 0). Therefore, the domain of the function is all points in the xy-plane except the origin.
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Discontinuity: The function has a discontinuity at the origin (0, 0). Let's see why:
- Undefined at (0, 0): The function is not defined at (0, 0) because it results in 0/0, an indeterminate form.
- Limit does not exist: As (x, y) approaches (0, 0) along different paths, the function's limit can take on different values. For instance:
- Approaching along the x-axis (y=0): lim (x, 0) → (0, 0) f(x, 0) = 1
- Approaching along the y-axis (x=0): lim (0, y) → (0, 0) f(0, y) = -1
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Continuity Elsewhere: Everywhere else in the xy-plane, the function is continuous. The function is a rational function, and rational functions are continuous wherever they are defined.
Conclusion
The function (x^2 - y^2) / (x^2 + y^2) is continuous for all points in the xy-plane except the origin (0, 0). At the origin, the function has a discontinuity because it is undefined and its limit does not exist.