https://file.aiquickdraw.com/mathbot/file/1728392874115509xgeri.jpg

The problem in the image is asking you to analyze the behavior of the function:

\[
f(x, y) = \frac{x^3 y^3}{x^2 + y^4}
\]

Specifically, it asks you to:
1. Show that \( f(x, y) \) tends to 0 along any curve of the form \( y = x^m \) for \( m \in \mathbb{N} \setminus \{3\} \) (where \( m \) is any natural number except 3).
2. Show that \( f(x, y) \) does *not* tend to 0 along the curve \( y = x^3 \).
3. Discuss what these results imply about the limit \( \lim_{(x, y) \to (0, 0)} f(x, y) \).

### Step 1: Check the limit along \( y = x^m \) for \( m \neq 3 \)

Substitute \( y = x^m \) into the expression for \( f(x, y) \):

\[
f(x, x^m) = \frac{x^3 (x^m)^3}{x^2 + (x^m)^4} = \frac{x^3 x^{3m}}{x^2 + x^{4m}} = \frac{x^{3 + 3m}}{x^2 + x^{4m}}.
\]

Now, analyze the behavior as \( x \to 0 \). The leading term in the denominator will depend on \( m \):
- If \( m > \frac{1}{2} \), then \( x^{4m} \) dominates in the denominator as \( x \to 0 \).
- If \( m < \frac{1}{2} \), then \( x^2 \) dominates.

In all these cases except \( m = 3 \), as \( x \to 0 \), the numerator approaches 0 faster than the denominator. Hence, \( f(x, x^m) \to 0 \) for any \( m \neq 3 \).

### Step 2: Check the limit along \( y = x^3 \)

Now substitute \( y = x^3 \) into the function:

\[
f(x, x^3) = \frac{x^3 (x^3)^3}{x^2 + (x^3)^4} = \frac{x^3 x^9}{x^2 + x^{12}} = \frac{x^{12}}{x^2 + x^{12}}.
\]

As \( x \to 0 \), the term \( x^2 \) dominates the denominator (since \( x^{12} \) vanishes faster than \( x^2 \)). Thus:

\[
f(x, x^3) = \frac{x^{12}}{x^2 (1 + x^{10})} \to \frac{0}{0 + 1} = 0.
\]

### Step 3: Conclusion on the limit \( \lim_{(x, y) \to (0, 0)} f(x, y) \)

Since \( f(x, y) \) tends to different values (0 along some curves and does not approach 0 uniformly), we conclude that:

\[
\lim_{(x, y) \to (0, 0)} f(x, y)
\]

does *not* exist. The behavior of the function depends on the path taken toward the origin.

Would you like more details or have any further questions?

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Here are some related questions for further practice:

1. What happens to the limit of \( f(x, y) \) along the curve \( y = x^4 \)?
2. Can you verify the limit along \( y = x^m \) for negative values of \( m \)?
3. How would the function behave if the denominator had different powers of \( x \) and \( y \)?
4. What can you infer about the general behavior of rational functions like this one near singular points?
5. How does this analysis extend to polar coordinates?

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**Tip**: When evaluating multivariable limits, always try several paths to check if the limit is path-dependent or independent.

Show that f(x, y) = (x^3 * y^3) / (x^2 + y^4) tends to 0 along any curve of the form y = x^m for m ∈ ℕ \ {3}, but that it does not tend to zero along the curve y = x^3. What does that tell us about lim (x,y) → (0,0) f(x, y)?

This problem explores the behavior of the function f(x, y) = (x^3 * y^3) / (x^2 + y^4) along different curves and discusses the existence of the limit lim (x,y) → (0,0). Suitable for students studying Multivariable Calculus, this problem highlights the importance of path-dependency in limits.

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