Question

$ \lim_{x \to -\frac{3}{2}} \frac{(4x^2 - 6x)(4x^2 + 6x + 9)}{\sqrt{2x - \sqrt{3}}} $

• A. \(\frac{3\sqrt{17}}{2}\)
• B. \(\frac{3\sqrt{16}}{2}\)
• C. \(\frac{3\sqrt{15}}{2}\)
• D. \(\frac{3\sqrt{14}}{2}\)

Hint:

When dealing with limits that lead to indeterminate forms, consider factorizing, simplifying expressions, or using L'Hôpital's Rule to find a clear path to the solution.

Answer 1

The Correct Option is A

- First, factorize the quadratic expressions in the numerator if possible to simplify the expression. 
- Substitute \(x = -\frac{3}{2}\) in the simplified equation and check the value of the denominator, as \(x\) approaches \(-\frac{3}{2}\), to determine the behavior of the function. 
- If the function presents an indeterminate form like \( \frac{0}{0} \), apply L'Hôpital's Rule or algebraic manipulation to resolve the indeterminacy. 
- Finally, evaluate the limit to find the exact value.