Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\left(x^{2}+y^{2}\right)^{2}}\)

Step-by-Step Solution

Answer

The limit does not exist because it depends on the path taken to approach the origin.

Step 1: Analyze the limit direction

Evaluate the behavior of the function as we approach the origin along different paths. For example, consider approaching along the x-axis by setting \(y = 0\):\[ f(x, 0) = \frac{x \cdot 0}{(x^2 + 0^2)^2} = 0\]Similarly, along the y-axis \(x = 0\):\[ f(0, y) = \frac{0 \cdot y}{(0^2 + y^2)^2} = 0\]In both cases, the limit is 0.

Step 2: Consider another path approach

Next, consider approaching along the line \(y = x\):\[ f(x, x) = \frac{x \cdot x}{(x^2 + x^2)^2} = \frac{x^2}{(2x^2)^2} = \frac{x^2}{4x^4} = \frac{1}{4x^2}\]As \(x \to 0\), \(\frac{1}{4x^2}\) approaches \(\infty\). Thus, the limit along this path doesn't yield 0, indicating different limits for different paths.

Step 3: Conclusion on limit existence

Since the limits along \(y = 0\) and \(x = 0\) both led to 0, but the limit along \(y = x\) approaches \(\infty\), the overall limit does not exist. The function's limit depends on the path taken, indicating path dependence and non-existence of a unique limit at \((0,0)\).

Key Concepts

Limit Existence

Determining whether a multivariable limit exists can be a bit tricky. Unlike single-variable calculus, where the path to a limit is straightforward, multivariable calculus requires us to examine how a function behaves as we approach a point from various directions. To establish the existence of a limit, we need all possible paths leading to that point to yield the same limit value. If we come upon even one path that doesn't align with others, the limit is deemed non-existent.

For our example, we considered paths along the x-axis and y-axis, both resulting in a limit of 0. However, when we changed our approach to the line where \( y = x \), the limit was different, leading to the conclusion that the overall limit does not exist. This shows us the different outcomes that can arise based on the approach direction, highlighting the necessity to evaluate multiple paths when assessing limit existence.

Path Dependence

The concept of path dependence is pivotal in understanding multivariable limits. In essence, path dependence refers to how a function evaluates a limit based on the path taken as it approaches a particular point. If a function's limit varies with the path, it indicates path dependence and non-existence of the limit in the multivariable context.

In our example function \(\frac{x y}{(x^2 + y^2)^2}\), we discovered path-dependent behavior by approaching the origin from different directions; results along the x-axis and y-axis were consistent, giving a 0 limit. However, a different result emerged when approaching along \(y = x\), leading to an infinite limit. This discrepancy flags path dependence, emphasizing the importance of examining multiple paths in such scenarios.

Approaching Origin

Approaching a point in multivariable calculus, especially the origin, involves considering an infinite number of potential paths. Each path can potentially lead to a different outcome for the limit, necessitating a broader analysis than in single-variable calculus.

In our problem, as \((x, y)\) approaches \((0, 0)\), the path matters significantly. The origin is a special point where different coordinate directions can drastically affect a function's behavior. We examined straightforward axes, and a diagonal approach \(y = x\). Depending on the chosen paths, the results varied, making it evident that analyzing how a function approaches a fundamental point such as the origin is crucial for correct limit evaluation in multivariable scenarios.