To evaluate the limit:

$\lim_{x \to 6} \left( 2x^2 - 18x + 36 - x^3 - 3x^2 + 54x \right)$

Step 1: Simplify the expression inside the limit.

First, combine like terms:

$2x^2 - 18x + 36 - x^3 - 3x^2 + 54x$

Group the terms:

$= (-x^3) + (2x^2 - 3x^2) + (-18x + 54x) + 36$

Simplify each group:

$= -x^3 - x^2 + 36x + 36$

Step 2: Substitute $x = 6$ into the simplified expression.

Now substitute $x = 6$ into the expression:

$- (6)^3 - (6)^2 + 36(6) + 36$

Calculate each term:

$- (6)^3 = -216$ $- (6)^2 = -36$ $36(6) = 216$

So, the expression becomes:

$-216 - 36 + 216 + 36$

Step 3: Simplify the result.

$-216 + 216 = 0$ $-36 + 36 = 0$

Thus, the final result is:

$0 + 0 = 0$

Final Answer:

$\lim_{x \to 6} \left( 2x^2 - 18x + 36 - x^3 - 3x^2 + 54x \right) = 0$

Let me know if you'd like more details or have further questions! Here are some related questions to deepen your understanding:

1. What happens to the limit if the expression includes higher powers of $x$?
2. How would the result change if we were dealing with a rational expression instead?
3. What is the significance of the limit of a polynomial function as $x$ approaches a number?
4. How do limits help in finding derivatives of functions?
5. What are some common methods to evaluate limits for indeterminate forms like $\frac{0}{0}$?

Tip: When evaluating limits for polynomial expressions, simply substitute the value of $x$ directly if the expression is continuous.