The expression you've given appears to be a limit problem:

$\lim_{h \to 0} \frac{8\left(\left(\frac{1}{2} + h\right)^2 - \left(\frac{1}{2}\right)^2\right)}{h}$

We will simplify this step by step:

Step 1: Simplify the squared terms

First, expand the squared terms inside the parentheses:

$\left(\frac{1}{2} + h\right)^2 = \left(\frac{1}{2}\right)^2 + 2\left(\frac{1}{2}\right)h + h^2 = \frac{1}{4} + h + h^2$

So, we have:

$\lim_{h \to 0} \frac{8\left(\frac{1}{4} + h + h^2 - \frac{1}{4}\right)}{h}$

Step 2: Simplify the expression

Now, cancel out $\frac{1}{4}$ terms:

$\lim_{h \to 0} \frac{8\left(h + h^2\right)}{h}$

Step 3: Factor and simplify

Factor out $h$ from the numerator:

$\lim_{h \to 0} \frac{8h(1 + h)}{h}$

Now, cancel out the $h$ in the numerator and denominator:

$\lim_{h \to 0} 8(1 + h)$

Step 4: Apply the limit

Now, substitute $h = 0$ into the expression:

$8(1 + 0) = 8$

Final Answer:

The value of the limit is:

$\boxed{8}$

Would you like further details, or do you have any questions?

Here are 5 related questions for further exploration:

1. How do you compute the limit of a polynomial function as $h$ approaches 0?
2. What are the typical methods for simplifying limit expressions involving fractions?
3. How does factoring help simplify limit problems in calculus?
4. What is the difference between limits involving powers and those involving trigonometric functions?
5. How does the derivative definition relate to this type of limit problem?

Tip: For limit problems, always try to simplify the expression algebraically before applying the limit directly!