https://file.aiquickdraw.com/mathbot/file/17293065063117hgjgp0h.jpg

The limit problem in the image is as follows:

\[
\lim_{(x, y) \to (0,0)} \frac{x \cdot y^4}{x^4 + y^4}
\]

To evaluate this limit, we need to check if it tends to the same value along different paths as \( (x, y) \) approaches \( (0, 0) \). If the limit depends on the path, the overall limit does not exist.

### Step 1: Evaluate the limit along \( y = 0 \)
If \( y = 0 \), the expression becomes:
\[
\frac{x \cdot 0^4}{x^4 + 0^4} = 0
\]
So, along the path \( y = 0 \), the limit is 0.

### Step 2: Evaluate the limit along \( x = 0 \)
If \( x = 0 \), the expression becomes:
\[
\frac{0 \cdot y^4}{0^4 + y^4} = 0
\]
So, along the path \( x = 0 \), the limit is also 0.

### Step 3: Evaluate the limit along \( y = x \)
If we substitute \( y = x \), the expression becomes:
\[
\frac{x \cdot x^4}{x^4 + x^4} = \frac{x^5}{2x^4} = \frac{x}{2}
\]
As \( x \to 0 \), \( \frac{x}{2} \to 0 \).

### Step 4: Evaluate the limit along \( y = -x \)
If we substitute \( y = -x \), the expression becomes:
\[
\frac{x \cdot (-x)^4}{x^4 + (-x)^4} = \frac{x \cdot x^4}{x^4 + x^4} = \frac{x^5}{2x^4} = \frac{x}{2}
\]
Again, as \( x \to 0 \), \( \frac{x}{2} \to 0 \).

### Conclusion:
Since the limit along multiple paths (e.g., \( y = 0 \), \( x = 0 \), \( y = x \), and \( y = -x \)) all lead to the same result of 0, we conclude that the limit exists and is equal to 0.

Thus, the limit is:
\[
\boxed{0}
\]

Would you like more details on any of the steps? 

Here are 5 related questions:
1. How do you evaluate limits in multivariable calculus using polar coordinates?
2. What are the common strategies to prove a limit does not exist?
3. How does the limit change when evaluated along different types of curves like \( y = x^2 \)?
4. How would this limit behave if the exponents in the denominator were different?
5. What are some common pitfalls when evaluating limits in two variables?

**Tip:** When solving limits involving two variables, always check the behavior along multiple paths to ensure the limit is consistent.

Find the limit of (x*y^4) / (x^4 + y^4) as (x, y) approaches (0, 0).

Learn how to evaluate the limit of the multivariable function (x*y^4) / (x^4 + y^4) as (x, y) approaches (0,0). This problem covers key concepts in multivariable calculus, including path dependency and the evaluation of limits using multiple approaches. Suitable for university students in Calculus II or Multivariable Calculus.

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