The given limit expression is:

$\lim_{{h \to 0}} \frac{{(3 + h)^3 - 27}}{h}$

This is a classic limit problem, which involves the difference quotient and appears to be connected to the derivative of a function. Specifically, it resembles the definition of the derivative of $f(x) = x^3$ at $x = 3$.

Step-by-Step Explanation:

1. Expand $(3 + h)^3$: $(3 + h)^3 = 3^3 + 3 \cdot 3^2 \cdot h + 3 \cdot 3 \cdot h^2 + h^3$ Simplifying this: $(3 + h)^3 = 27 + 27h + 9h^2 + h^3$

2. Substitute into the original limit: The expression becomes: $\frac{{(27 + 27h + 9h^2 + h^3) - 27}}{h}$ Simplify the numerator: $= \frac{{27h + 9h^2 + h^3}}{h}$

3. Simplify by factoring out $h$: Factor $h$ from the numerator: $= \frac{{h(27 + 9h + h^2)}}{h}$ Cancel out $h$ (as long as $h \neq 0$): $= 27 + 9h + h^2$

4. Take the limit as $h \to 0$: Now, substitute $h = 0$ into the simplified expression: $27 + 9(0) + 0^2 = 27$

Thus, the value of the limit is:

$\boxed{27}$

Conclusion:

The limit evaluates to $27$, which is the derivative of $f(x) = x^3$ at $x = 3$.

Do you want more details on any specific part of this solution or have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the definition of the derivative using limits?
2. How would the approach change if the power were different, such as $(3 + h)^4$?
3. How is this limit problem related to the concept of instantaneous rate of change?
4. How can you use the binomial expansion to help solve similar limit problems?
5. What happens if the limit fails to exist for certain functions?

Tip: When dealing with limits that resemble derivative definitions, identifying the underlying function can help simplify the problem greatly.