The expression given is:

$\lim_{x \to 11.5} \left( 4x^2 + 8x - 6 \right)^{\frac{3}{2}}$

To evaluate this limit, we can follow these steps:

1. Substitute $x = 11.5$ directly into the expression $4x^2 + 8x - 6$.
2. Compute the expression inside the parentheses.
3. Raise the result to the power of $\frac{3}{2}$.

Let's calculate it step by step.

Step 1: Substitute $x = 11.5$ into the expression

$4(11.5)^2 + 8(11.5) - 6$

Step 2: Calculate the expression inside the parentheses

$4(132.25) + 92 - 6 = 529 + 92 - 6 = 615$

Step 3: Raise the result to the power of $\frac{3}{2}$

$615^{\frac{3}{2}} = (615^{\frac{1}{2}})^3 = (\sqrt{615})^3$

The square root of 615 is approximately 24.8, so:

$(24.8)^3 \approx 15252.35$

Final Answer:

$\lim_{x \to 11.5} \left( 4x^2 + 8x - 6 \right)^{\frac{3}{2}} = 15252.35$

Would you like a more detailed explanation or have any further questions?

Here are five related questions for further exploration:

1. How do we compute limits involving polynomials?
2. What is the significance of the exponent $\frac{3}{2}$ in this problem?
3. How do square roots affect the calculation of limits?
4. Can limits be evaluated for undefined expressions?
5. How do you handle limits that approach infinity?

Tip: When evaluating limits, direct substitution is often the first and simplest method to try, especially if the function is continuous at the point of interest.