The image presents the following problem:

Given: $\lim_{x \to \infty} f(x) = -9, \quad \lim_{x \to \infty} g(x) = 8$ Evaluate: $\lim_{x \to \infty} \frac{x^2 f(x) + 8x - 10}{x^2 g(x) + 4x - 6}$

Solution:

1. Start by analyzing the limits of each term in both the numerator and the denominator as $x \to \infty$.

• Numerator: $x^2 f(x) + 8x - 10 = x^2 (-9) + 8x - 10 = -9x^2 + 8x - 10$ As $x \to \infty$, the dominant term in the numerator is $-9x^2$.

• Denominator: $x^2 g(x) + 4x - 6 = x^2 (8) + 4x - 6 = 8x^2 + 4x - 6$ As $x \to \infty$, the dominant term in the denominator is $8x^2$.