The problem is asking to evaluate the limit:

$\lim_{x \to \frac{15}{2}} \left( 4x^2 + 6x - 6 \right)^{\frac{3}{2}}$

To solve this, we can directly substitute $x = \frac{15}{2}$ into the expression inside the parentheses, following the limit properties provided:

$\lim_{x \to c} \left( f(x) \right)^{\frac{p}{q}} = \left( \lim_{x \to c} f(x) \right)^{\frac{p}{q}}$

Now, we substitute $x = \frac{15}{2}$ into the expression $4x^2 + 6x - 6$.

Step 1: Substitute $x = \frac{15}{2}$

$4\left( \frac{15}{2} \right)^2 + 6\left( \frac{15}{2} \right) - 6$

$= 4 \times \frac{225}{4} + 6 \times \frac{15}{2} - 6$

$= 225 + 45 - 6$

$= 264$

Step 2: Take the exponent $\frac{3}{2}$ of 264

$(264)^{\frac{3}{2}} = \sqrt{264^3}$

Therefore, the expression simplifies to:

$\sqrt{264}$

The answer provided in the image is correct.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you evaluate limits involving square roots and exponents?
2. Can you apply L'Hopital's Rule for indeterminate forms in limits?
3. What are the rules for simplifying expressions with fractional exponents?
4. How do polynomial functions behave near limits?
5. What are other methods for solving limits without substitution?

Tip:

Always check for any indeterminate forms (like $\frac{0}{0}$) before applying substitution in limits.