ㄹ. \( \lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+1} \)
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To solve the limit \( \lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+1} \), we start by analyzing the expression as \( x \) approaches negative infinity. First, simplify the expression inside the square root and the denominator: \[ \sqrt{9 x^6 - x} \text{ can be approximated by } \sqrt{9 x^6} = 3|x^3| = 3(-x^3) = -3x^3 \text{ for large negative } x. \] The denominator behaves as follows: \[ x^3 + 1 \text{ approaches } x^3 \text{ for large negative } x. \] Now, substituting these simplifications into the limit, we have: \[ \lim_{x \rightarrow -\infty} \frac{\sqrt{9 x^6 - x}}{x^3 + 1} \approx \lim_{x \rightarrow -\infty} \frac{-3 x^3}{x^3} = \lim_{x \rightarrow -\infty} -3. \] Thus, the limit evaluates to: \[ \boxed{-3}. \]
