To evaluate the limit

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

we can directly substitute $x = \frac{3}{2}$ into the expression because the function inside the limit is continuous. Let's compute this step by step.

Step 1: Substitute $x = \frac{3}{2}$ into the quadratic expression

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

Step 2: Compute $\left(\frac{3}{2}\right)^2$

$\left(\frac{3}{2}\right)^2 = \frac{9}{4}$

Step 3: Multiply by 4

$4 \times \frac{9}{4} = 9$

Step 4: Compute $8 \times \frac{3}{2}$

$8 \times \frac{3}{2} = 12$

Step 5: Add the results and subtract 6

$9 + 12 - 6 = 15$

Step 6: Raise the result to the power of $\frac{3}{2}$

$15^{3/2}$

Step 7: Simplify $15^{3/2}$

This is equivalent to taking the square root of 15 and then cubing the result:

$\sqrt{15} \approx 3.87$

Then:

$3.87^3 \approx 58.08$

So,

$\lim_{x \to \frac{3}{2}} \left(4x^2 + 8x - 6\right)^{3/2} \approx 58.08.$

This is the approximate value of the limit.

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Would you like more details, or do you have any questions? Here are some related questions you might find useful:

1. What are the rules for computing limits involving fractional exponents?
2. How does continuity affect the evaluation of limits?
3. What methods can be used if direct substitution is not possible?
4. How can you approximate irrational numbers like $\sqrt{15}$ when solving limits?
5. What are some common mistakes to avoid when evaluating limits with complex expressions?

Tip: When dealing with expressions involving fractional exponents, remember to simplify the base first before applying the exponent.