To find the value that the function $f(x, y)$ approaches as $(x, y) \to (0, 0)$ along the $y$-axis, let's analyze the behavior of the function when $x = 0$. This corresponds to approaching the origin along the $y$-axis.

The function is given by:

$f(x, y) = \frac{y^4 - 2x^2}{y^4 + x^2}$

Along the $y$-axis, we have $x = 0$, so the function simplifies to:

$f(0, y) = \frac{y^4 - 2(0)^2}{y^4 + (0)^2} = \frac{y^4}{y^4} = 1$

Therefore, as $(x, y) \to (0, 0)$ along the $y$-axis, the value of the function approaches $1$.

Conclusion:

The correct answer is:

A. $f(x, y)$ approaches $1$.

Let me know if you'd like further details or if you have any other questions!

Here are some related questions for you:

1. How do we determine limits for functions of two variables?
2. What happens when you approach a point along different paths (e.g., along the $x$-axis)?
3. Can a limit in two variables be path-dependent?
4. How do we handle cases where the denominator approaches zero in a two-variable limit?
5. What does it mean if a limit does not exist as $(x, y) \to (0, 0)$?

Tip: When analyzing a two-variable limit, always try simplifying the expression along different paths to check for path dependence.