The given limit problem is:

$\lim_{x \to 3} \left(-x^3 - 4x^2 + x - 3\right)$

To solve this, we can directly substitute $x = 3$ into the expression because it's a polynomial function, which is continuous everywhere.

Let’s substitute $x = 3$:

$- (3^3) - 4(3^2) + 3 - 3$

Simplifying:

$- 27 - 4(9) + 3 - 3 = -27 - 36 + 3 - 3$

$= -27 - 36 = -63$

Thus, the value of the limit is:

$\boxed{-63}$

Would you like a more detailed explanation or have any questions? Here are some related questions for further practice:

1. What is the general method for solving limits of polynomial functions?
2. How can you determine when a function has a continuous limit at a certain point?
3. What happens if the function has a discontinuity at the point where you're evaluating the limit?
4. How do you apply L'Hopital's Rule to limits with indeterminate forms?
5. What are the properties of limits that make solving them easier?

Tip: For polynomials, limits as $x$ approaches any real number can often be solved simply by direct substitution, since polynomials are continuous everywhere.