Evaluate the limit \( \lim _{x \rightarrow \infty} \frac{10 x^{3}-3 x^{2}-11 x}{9-11 x-4 x^{3}} \)

To evaluate the limit \[ \lim_{x \rightarrow \infty} \frac{10x^3 - 3x^2 - 11x}{9 - 11x - 4x^3}, \] we can analyze the behavior of the polynomial expressions as \( x \) approaches infinity. ### Step 1: Identify the Leading Terms In both the numerator and the denominator, the highest power of \( x \) will dominate the behavior of the expression as \( x \) becomes very large. - **Numerator:** The leading term is \( 10x^3 \). - **Denominator:** The leading term is \( -4x^3 \). ### Step 2: Simplify by Dividing by \( x^3 \) To make the comparison clearer, divide both the numerator and the denominator by \( x^3 \): \[ \frac{10x^3 - 3x^2 - 11x}{9 - 11x - 4x^3} = \frac{10 - \frac{3}{x} - \frac{11}{x^2}}{\frac{9}{x^3} - \frac{11}{x^2} - 4}. \] ### Step 3: Evaluate the Limit As \( x \) approaches infinity: - \( \frac{3}{x} \), \( \frac{11}{x^2} \), and \( \frac{9}{x^3} \) all approach 0. So, the expression simplifies to: \[ \frac{10}{-4} = -\frac{10}{4} = -\frac{5}{2}. \] ### Conclusion The limit is: \[ -\frac{5}{2}. \] **Answer:** \(-\dfrac{5}{2}\)