It seems like the text extraction didn't capture the mathematical expression correctly. However, from the image, the limit appears to be:

$\lim_{{h \to 0}} \frac{(2+h)^3 + 2(2+h) - 2^3 - 2(2)}{h}$

Let's solve this limit step-by-step.

Step 1: Expand $(2+h)^3$

$(2+h)^3 = 8 + 12h + 6h^2 + h^3$

Step 2: Substitute the expanded form into the expression

The expression becomes:

$\frac{(8 + 12h + 6h^2 + h^3) + 2(2+h) - 8 - 4}{h}$

Step 3: Simplify the terms

Expand and combine the terms: $\frac{(8 + 12h + 6h^2 + h^3) + 4 + 2h - 8 - 4}{h} = \frac{12h + 6h^2 + h^3 + 2h}{h}$

Step 4: Factor the numerator

$= \frac{h(12 + 6h + h^2 + 2)}{h} = 12 + 6h + h^2 + 2$

Step 5: Take the limit as $h \to 0$

$\lim_{{h \to 0}} (12 + 6h + h^2 + 2) = 12 + 2 = 14$

Thus, the value of the limit is:

$\boxed{14}$

Let me know if you need more details or further clarifications!

Here are 5 related questions to explore:

1. How do you calculate limits involving indeterminate forms?
2. What is the binomial expansion and how is it applied?
3. How do you handle limits with higher-order polynomials?
4. Can L'Hopital's Rule be used in this problem, and if so, how?
5. What are common techniques to simplify limit problems involving polynomials?

Tip: Always check if you can simplify the numerator and denominator before applying limit techniques!