To evaluate the limit

$\lim_{x \to 232} \left(4x^2 + 8x - 6\right)^{3/2},$

we can directly substitute $x = 232$ into the expression because the function inside the limit is continuous. Let's compute this step by step.

Step 1: Substitute $x = 232$ into the quadratic expression

$4x^2 + 8x - 6 = 4(232)^2 + 8(232) - 6$

Step 2: Compute $232^2$

$232^2 = 53824$

Step 3: Multiply by 4

$4 \times 53824 = 215296$

Step 4: Compute $8 \times 232$

$8 \times 232 = 1856$

Step 5: Add the results and subtract 6

$215296 + 1856 - 6 = 217146$

Step 6: Raise the result to the power of $3/2$

$\text{Final result} = (217146)^{3/2}$

Step 7: Compute $217146^{3/2}$

This is equivalent to taking the square root of 217146 and then cubing the result:

$\sqrt{217146} \approx 466 \quad \text{(approximation)}$

Then:

$466^3 = 101194696$

So,

$\lim_{x \to 232} \left(4x^2 + 8x - 6\right)^{3/2} \approx 101194696.$

This is the 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 is the meaning of continuity in the context of limits?
2. How do you handle limits that approach infinity or negative infinity?
3. What are some techniques for evaluating limits without direct substitution?
4. How does the power rule work when dealing with limits?
5. What is the significance of the exponent $\frac{3}{2}$ in this context?

Tip: When dealing with limits, always check if the function is continuous at the point of interest, as this allows for direct substitution to find the limit.