Question 9 Evaluate the limit: limx→−∞2x3+x2−5x3−4x+1\lim_{x \to -\infty} \frac{2x^3 + x^2 - 5}{x^3 - 4x + 1} Show solutionHide solution+Question 9 - Solution We are given: limx→−∞2x3+x2−5x3−4x+1\lim_{x \to -\infty} \frac{2x^3 + x^2 - 5}{x^3 - 4x + 1} Step 1: Factor out the highest power of xx (which is x3x^3) from both numerator and denominator: =limx→−∞x3(2+1x−5x3)x3(1−4x2+1x3)= \lim_{x \to -\infty} \frac{x^3 \left(2 + \frac{1}{x} - \frac{5}{x^3} \right)}{x^3 \left(1 - \frac{4}{x^2} + \frac{1}{x^3} \right)} Cancel the common x3x^3 terms: =limx→−∞2+1x−5x31−4x2+1x3= \lim_{x \to -\infty} \frac{2 + \frac{1}{x} - \frac{5}{x^3}}{1 - \frac{4}{x^2} + \frac{1}{x^3}} Step 2: Take the limit as x→−∞x \to -\infty. All terms with 1xn→0\frac{1}{x^n} \to 0: =2+0−01−0+0=2= \frac{2 + 0 - 0}{1 - 0 + 0} = \boxed{2}
