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Limits in calculus are fundamental in understanding the behavior of functions as variables approach a certain point. They help us evaluate how a function behaves around a given value, which is crucial in both single-variable and multivariable calculus. When dealing with limits, you essentially check the value that a function approaches as its input gets closer to some specified point. Here, we're considering a limit problem that involves two variables, \(x\) and \(y\).

This means that instead of just approaching a point on the line, we approach a point on a plane, which adds complexity. Calculating limits in this context helps determine whether we can assume continuity or whether the function behaves nicely around the point of interest. In the problem given, we are asked to determine if the limit exists as \((x, y)\) approaches \((1, 1)\). When exploring multivariable limits, it's essential to not only approach from both axes but also from different directions on the plane.

This exploration may reveal that the function behaves differently based on the path taken, hinting at path-dependence and potentially indicating non-existence of a limit.

Path dependence is a crucial idea when evaluating multivariable limits. Simply put, a limit is path-dependent when its value changes depending on the route taken to approach it. In single-variable calculus, you're usually limited to two directions (from the left or from the right), making path dependence less of an issue. However, with multivariable calculus, the paths are infinite, as you can approach the point from numerous curves or lines.

In the exercise provided, checking the limit along different paths, such as \(x = 1\) and \(y = x\), showed different resulting values, namely 2 and 3. The path along \(x = 1\) yielded a limit of 2, while approaching the point with \(y = x\) led to a limit of 3. The fact that these two paths give us different results is a clear indication of path dependence. A true multivariable limit should give the same value no matter the path chosen.

Therefore, when evaluating these limits, it's crucial to test multiple paths. If any path gives a different limit value, it signals that the overall limit does not exist.

Non-existent limits occur when a function doesn't approach a single value as its inputs converge to a specific point. This often happens because the function behaves differently based on how you approach that point. In the multivariable context, this is closely linked to path dependence.

For example, in the provided exercise, the limit of the function as \((x, y)\) heads towards \((1, 1)\) was examined. Testing different paths showed distinct outcomes, leading to the conclusion that the limit does not exist. This happens because a valid limit in multivariable calculus implies convergence to the same value across all potential paths. If even one path diverges or yields a different value, then the limit cannot be said to exist.

Thus, non-existent limits remind us of the complexities in understanding functions of several variables and highlight the importance of analyzing behavior from multiple angles to fully grasp the limit's existence or lack thereof.