The given problem is:

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

This limit can be evaluated using the definition of a derivative or Taylor expansion. Let’s proceed step-by-step:

Step 1: Recognize the structure

This is in the form of a derivative. Specifically, the numerator resembles $f(a + h) - f(a)$, where: $f(x) = 8x^8 \quad \text{and} \quad a = \frac{1}{2}.$

Thus, the expression becomes: $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}.$

This is the definition of the derivative $f'(a)$.

Step 2: Compute the derivative of $f(x)$

$f(x) = 8x^8.$ Differentiating: $f'(x) = 8 \cdot 8x^{7} = 64x^7.$

Step 3: Evaluate $f'(a)$ at $a = \frac{1}{2}$

Substituting $x = \frac{1}{2}$ into $f'(x)$: $f' \left(\frac{1}{2}\right) = 64 \left(\frac{1}{2}\right)^7 = 64 \cdot \frac{1}{128}.$

Simplify: $f' \left(\frac{1}{2}\right) = \frac{64}{128} = \frac{1}{2}.$

Final Answer:

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

Would you like a detailed explanation of any step, or have other questions?

5 Related Questions:

1. How does the definition of a derivative apply to limits like this one?
2. Can this limit be solved using a binomial expansion instead of derivatives?
3. How do you find the derivative of functions with powers, e.g., $x^n$?
4. What are the key rules for simplifying derivatives involving constants like $8x^8$?
5. How would this problem change if $f(x)$ were $8x^{10}$ instead of $8x^8$?

1 Tip:

When solving limits involving powers or differences like $f(a + h) - f(a)$, look for opportunities to apply the derivative definition—it simplifies calculations!