91Ó°ÊÓ

Polar coordinates offer a way to express the position of a point in terms of a distance and an angle. In our situation, as \(x, y\) approaches \(0, 0\), we convert Cartesian coordinates to polar coordinates using \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Here, \(r\) is the radial distance from the origin, calculated as \(\sqrt{x^2 + y^2}\), and \(\theta\) is the angle with respect to the positive x-axis.

Converting to polar coordinates helps simplify the limit problem by reducing the dimensionality of the input. Instead of considering movement in two dimensions separately, polar coordinates allow us to view the problem in terms of radial direction and angular orientation. This transformation organizes the problem's behavior as \(r \ ightarrow 0\) and makes it easier to manipulate, often revealing whether or not a limit exists."

In our function, \(f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}\), converting to polar coordinates converts the expression to \(\cos(2\theta)\). This shows that the function does not depend on \(r\), hinting that the approach path's angle \(\theta\) plays a crucial role.

Path dependence occurs when the result of a function's limit varies depending on the path taken toward a particular point. In multivariable calculus, we often examine limits approaching \(0,0\) by taking various paths such as lines, curves, or even spirals. For our function \(f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}\), substitute \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\), simplifying the expression to \(\cos(2\theta)\). This result makes it evident that the final value is solely dependent on the angle \(\theta\), indicating strong path dependence.

Simply put, two different paths toward \(0,0\) can yield different values for \(f(x,y)\). For example:

• Along the x-axis: \(y = 0\), then \(\theta = 0\), so \(\cos(2 \times 0) = 1\).
• Along the y-axis: \(x = 0\), then \(\theta = \frac{\pi}{2}\), so \(\cos(2 \times \frac{\pi}{2}) = -1\).

Since varying paths lead to different values, arriving at a consistent limit value is impossible. This path dependence shows why the function does not have a well-defined limit.

When we say "the limit does not exist," it means there isn't a single value the function approaches as the input gets arbitrarily close to a particular point. The limit should be unique regardless of the approach path. For the given function \(f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}\), analyzing the expression in polar coordinates \(\cos(2\theta)\) shows that different paths yield different limiting values.

The central issue here is that each approach direction, corresponding to a unique angle \(\theta\), results in a different output for \(\cos(2\theta)\). Therefore, there is no single value the function converges to, making the limit undefined. The path dependency finding from our exploration into polar coordinates supports the conclusion that

• If a single limit existed, all paths reaching \(0,0\) would provide the same function value.
• Since they do not, the limit cannot be defined consistently.

So, for functions like these, if changing the path changes the result, the limit does not exist. It's a key finding about the behavior of multivariable functions.