The problem in the image appears to be about determining the limit of the following expression:

$\lim_{{x \to 2}} \frac{(3x - 1)^3 - (x + 3)^3}{x^3 - 8}$

Solution Steps:

1. Direct Substitution (Substitusi Langsung):

   • First, substituting $x = 2$ directly into the expression gives:

   $\lim_{{x \to 2}} \frac{(3(2) - 1)^3 - (2 + 3)^3}{2^3 - 8} = \frac{(5)^3 - (5)^3}{8 - 8} = \frac{125 - 125}{0} = \frac{0}{0}$

   This results in an indeterminate form $\frac{0}{0}$, so we need to manipulate the expression further.

2. Factoring the Expression:

   • To resolve the indeterminate form, the expression can be factored.
   • Expanding $(3x - 1)^3$ and $(x + 3)^3$ gives:

   $(3x - 1)^3 = 27x^3 - 27x^2 + 9x - 1$ $(x + 3)^3 = x^3 + 9x^2 + 27x + 27$

   So, the numerator becomes:

   $(27x^3 - 27x^2 + 9x - 1) - (x^3 + 9x^2 + 27x + 27) = 26x^3 - 36x^2 - 18x - 28$

   Factoring this expression out of common terms:

   $2(x - 2)(13x^2 + 8x + 7)$

3. Simplifying the Denominator:

   • The denominator $x^3 - 8$ can be factored as:

   $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$

4. Canceling Common Terms:

   • Now, we can cancel $(x - 2)$ from both the numerator and denominator:

   $\lim_{{x \to 2}} \frac{2(x - 2)(13x^2 + 8x + 7)}{(x - 2)(x^2 + 2x + 4)} = \lim_{{x \to 2}} \frac{2(13x^2 + 8x + 7)}{x^2 + 2x + 4}$

5. Substitute $x = 2$ Again:

   • Now, substitute $x = 2$ into the simplified expression:

   $\frac{2(13(2)^2 + 8(2) + 7)}{(2)^2 + 2(2) + 4} = \frac{2(13(4) + 16 + 7)}{4 + 4 + 4} = \frac{2(52 + 16 + 7)}{12} = \frac{2(75)}{12} = \frac{150}{12} = \frac{25}{2}$

Thus, the final value of the limit is:

$\boxed{\frac{25}{2}}$

Would you like more details on any step of the process?

Here are five questions to further your understanding:

1. How do you identify indeterminate forms like $\frac{0}{0}$?
2. What are some alternative ways to solve limits aside from factoring?
3. Can L'Hopital's Rule be applied in this problem? If so, how?
4. Why is factoring an effective method for solving limits involving indeterminate forms?
5. How can you verify the result of this limit calculation using graphical or numerical methods?

Tip: Always attempt direct substitution first when evaluating limits—it quickly shows if further manipulation is necessary.