The given function is:

$f(x, y) = \tan^{-1} \left( \frac{|x| + |y|}{x^2 + y^2} \right)$

To find the limit of this function as $(x, y) \to (0, 0)$, let's analyze it.

Step 1: Understanding the Behavior Near the Origin

As $(x, y) \to (0, 0)$, the function’s behavior is influenced by both the numerator $|x| + |y|$ and the denominator $x^2 + y^2$.

Step 2: Polar Coordinates Substitution

Convert the function to polar coordinates where $x = r\cos\theta$ and $y = r\sin\theta$. This transformation is useful as it simplifies expressions involving $\sqrt{x^2 + y^2}$.

$|x| + |y| = r(|\cos\theta| + |\sin\theta|)$

$x^2 + y^2 = r^2$

So, the function becomes:

$f(r, \theta) = \tan^{-1} \left( \frac{r(|\cos\theta| + |\sin\theta|)}{r^2} \right) = \tan^{-1} \left( \frac{|\cos\theta| + |\sin\theta|}{r} \right)$

Step 3: Limit Analysis as $r \to 0$

As $r \to 0$, the term $\frac{|\cos\theta| + |\sin\theta|}{r}$ grows without bound because the denominator $r$ approaches zero while the numerator remains finite. Specifically, $|\cos\theta| + |\sin\theta|$ is always positive and bounded between $0$ and $\sqrt{2}$.

Thus,

$\frac{|\cos\theta| + |\sin\theta|}{r} \to \infty \text{ as } r \to 0.$

Step 4: Behavior of $\tan^{-1}(x)$ as $x \to \infty$

$\tan^{-1}(x)$ approaches $\frac{\pi}{2}$ as $x \to \infty$.

Conclusion:

$\lim_{(x, y) \to (0, 0)} f(x, y) = \frac{\pi}{2}.$

Summary:

The limit of the function $f(x, y)$ as $(x, y) \to (0, 0)$ is $\frac{\pi}{2}$.

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Would you like more details on this solution, or do you have any questions? Here are some related questions you might find interesting:

1. How does the use of polar coordinates help in evaluating limits involving two variables?
2. What are other techniques to evaluate limits that involve absolute values and powers?
3. Can we use the Squeeze Theorem to find the limit of similar functions?
4. How does the behavior of $\tan^{-1}(x)$ affect limit evaluation?
5. What are other examples of functions where the limit is influenced heavily by polar transformations?

Tip: Converting to polar coordinates can greatly simplify limit evaluation in two dimensions, especially when dealing with circular or symmetric behavior around the origin.