Limits at Infinity, Part I — Question 3

Question 3 - Solution

We are given: $$\lim_{x \to \infty} \frac{3x^2 - x + 4}{5x^2 + 7x - 9}$$

To evaluate this limit, we divide both numerator and denominator by $x^2$, the highest power of $x$:

$$\lim_{x \to \infty} \frac{3 - \frac{1}{x} + \frac{4}{x^2}}{5 + \frac{7}{x} - \frac{9}{x^2}}$$

As $x \to \infty$, the terms $\frac{1}{x}, \frac{4}{x^2}, \frac{7}{x}, \frac{9}{x^2} \to 0$, so we get:

$$\frac{3 - 0 + 0}{5 + 0 - 0} = \frac{3}{5}$$

Final Answer: $$\boxed{\frac{3}{5}}$$