The limit provided in the image is:

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

To evaluate this limit, we need to check if it exists by applying an appropriate method. One typical method is to approach the origin $(0, 0)$ along different paths and check if the result is the same for all paths. If the limit depends on the path, then the limit does not exist.

Path 1: Along the x-axis (y = 0)

When $y = 0$, the expression becomes: $\frac{x^4 \cdot 0^4}{x^4 + 0^4} = \frac{0}{x^4} = 0$

So, along the x-axis, the limit is 0.

Path 2: Along the y-axis (x = 0)

When $x = 0$, the expression becomes: $\frac{0^4 \cdot y^4}{0^4 + y^4} = \frac{0}{y^4} = 0$

Along the y-axis, the limit is also 0.

Path 3: Along the line $y = x$

If we substitute $y = x$, the expression becomes: $\frac{x^4 \cdot x^4}{x^4 + x^4} = \frac{x^8}{2x^4} = \frac{x^4}{2}$ As $x \to 0$, this expression tends to $\frac{0^4}{2} = 0$.

Conclusion

Since the limit is 0 along all the paths we checked, we can conclude that:

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

Would you like more details or have any questions?

Here are five related questions you could explore:

1. How do limits behave when approaching points from different paths?
2. What happens when the numerator has a higher degree than the denominator in limits?
3. Can polar coordinates simplify the evaluation of limits in two dimensions?
4. What are some cases where limits do not exist due to path dependence?
5. How do L'Hopital's Rule or other techniques apply to multivariable limits?

Tip: Checking the limit along different paths is a powerful method for identifying whether a limit depends on the approach direction. If the limit varies between paths, it does not exist!