The function $f(x, y) = x^a y^b x^m + y^n$, defined on $\mathbb{R}^2 \setminus \{0\}$, where $m$ and $n$ are positive even numbers, and $a, b \in \mathbb{R}$, can be analyzed as $(x, y) \to (0, 0)$.

Step-by-Step Analysis

1. Understanding the Behavior Near the Origin: We are interested in how the function behaves as $(x, y)$ approaches the origin $(0, 0)$. To do so, we must examine the limits along different paths approaching the origin and check for consistency.

2. General Form of the Function: The function is given by: $f(x, y) = x^a y^b x^m + y^n$ which can be split into two terms:

   • The first term: $x^a y^b x^m = x^{a+m} y^b$
   • The second term: $y^n$

3. Limit Along Different Paths: To understand the limit behavior, it's useful to approach the origin along various straight lines and curves:

   • Along the x-axis (where $y = 0$): $f(x, 0) = x^{a+m} \cdot 0^b + 0^n = 0$ Thus, along the x-axis, $f(x, y) \to 0$ as $x \to 0$.

   • Along the y-axis (where $x = 0$): $f(0, y) = 0^{a+m} y^b + y^n = y^n$ Since $n$ is even, as $y \to 0$, $f(0, y) \to 0$.

   • Along a line $y = mx$ (for some constant $m$): Substituting $y = mx$ into the function: $f(x, mx) = x^{a+m} (mx)^b + (mx)^n$ This simplifies to: $f(x, mx) = m^b x^{a+m+b} + m^n x^n$ As $x \to 0$, the behavior of $f(x, mx)$ depends on the powers of $x$. Specifically:

     • If $a + m + b > n$, the term $m^b x^{a+m+b}$ will dominate, and $f(x, mx) \to 0$.
     • If $a + m + b < n$, the term $m^n x^n$ will dominate, and $f(x, mx) \to 0$.
     • If $a + m + b = n$, the limit depends on the constants involved but should approach zero as well.

4. Conclusion on Limit: The behavior of $f(x, y)$ as $(x, y) \to (0, 0)$ is generally 0 along various paths as long as $m$ and $n$ are positive even numbers. The function has a strong tendency to approach 0 along straight lines and curves, indicating that the limit is likely 0 as $(x, y) \to (0, 0)$.

   However, to formally prove this, one could use polar coordinates or other methods, but the above reasoning gives a solid indication that the limit is indeed 0 at the origin.

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Would you like to explore the behavior in more detail (e.g., using polar coordinates or another method), or do you have any questions about this analysis?

Related Questions:

1. How would the behavior of the function change if $m$ and $n$ were odd numbers instead of even?
2. What happens to the limit if we allow $x$ and $y$ to approach the origin along a curve other than a straight line?
3. How can polar coordinates be used to further confirm the behavior of this function at the origin?
4. What would happen to the function if the powers $a$ and $b$ were both negative?
5. How do the signs of $a$ and $b$ affect the limit behavior of the function?

Tip:

To study the limit in more general settings, it can be helpful to express the function in polar coordinates $(r, \theta)$, where $x = r \cos(\theta)$ and $y = r \sin(\theta)$, and analyze how the function behaves as $r \to 0$.