\[
\lim_{x \to -\frac{3}{2}} \frac{(4x^2 - 6x)(4x^2 + 6x + 9)}{\sqrt{2x - \sqrt{3}}}
\]
\[ \lim_{x \to -\frac{3}{2}} \frac{(4x^2 - 6x)(4x^2 + 6x + 9)}{\sqrt{2x - \sqrt{3}}} \]
Show Hint
- \(\frac{3\sqrt{17}}{2}\)
- \(\frac{3\sqrt{16}}{2}\)
- \(\frac{3\sqrt{15}}{2}\)
\(\frac{3\sqrt{14}}{2}\)
The Correct Option is A
Solution and Explanation
- 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.
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