Limits At Infinity, Part II — Question 9

Question 9 - Solution

We are given the rational function: $$\lim_{x \to -\infty} \frac{x^3 + 4x^2 - 7}{2x^3 - 5x + 1}$$

—

Step 1: Analyze degrees of numerator and denominator

- The highest degree in both numerator and denominator is 3. - For limits at infinity, when the degrees match, the limit is the ratio of the leading coefficients.

So: $$\lim_{x \to -\infty} \frac{x^3 + 4x^2 - 7}{2x^3 - 5x + 1} = \frac{1}{2}$$

—

Step 2: Justify with algebraic manipulation (optional)

Divide numerator and denominator by $x^3$: $$= \frac{1 + \frac{4}{x} - \frac{7}{x^3}}{2 - \frac{5}{x^2} + \frac{1}{x^3}}$$

As $x \to -\infty$, the fractions go to 0: $$\frac{1 + 0 - 0}{2 - 0 + 0} = \frac{1}{2}$$

—

Final Answer: $$\boxed{\frac{1}{2}}$$