lim*𝑥*→11.5(4*𝑥*2+8*𝑥*−6)3/2

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.

Learn how to evaluate the limit of (4x^2 + 8x - 6)^(3/2) as x approaches 11.5 with a detailed step-by-step solution. This problem covers limits involving polynomials and exponentiation rules. Suitable for students in Grades 11-12.

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