To find the limit of the function $4x^3 - 2xy^2$ as $(x, y)$ approaches $(-5, 4)$, we can substitute the values directly if the function is continuous at that point.

1. Substitute $x = -5$ and $y = 4$ into the function: $4(-5)^3 - 2(-5)(4)^2$

2. Calculate each term separately:

   • For the first term, $4(-5)^3$: $4 \times (-125) = -500$

   • For the second term, $-2(-5)(4)^2$: $-2 \times (-5) \times 16 = 2 \times 5 \times 16 = 160$

3. Combine the results: $-500 + 160 = -340$

Therefore, the limit is: $\lim_{(x,y) \to (-5,4)} 4x^3 - 2xy^2 = -340$

Would you like further clarification on any step, or do you have additional questions?

Here are five related questions that could deepen your understanding:

1. What does it mean for a function to be continuous at a point?
2. How do we handle limits when direct substitution leads to an undefined form?
3. What is the significance of approaching a limit in multivariable calculus?
4. How would we approach the problem if the limit did not exist?
5. How do partial derivatives relate to finding limits in multivariable calculus?

Tip: In multivariable calculus, always check if direct substitution works first for limits, as it can save time and simplify the solution.