The problem involves evaluating the limit as $x \to \infty$ of:

$\frac{(2x + 3)^2}{5 - 3x - x^2}.$

Step-by-step Solution:

1. Expand the numerator: $(2x + 3)^2 = 4x^2 + 12x + 9.$

   So, the fraction becomes: $\frac{4x^2 + 12x + 9}{5 - 3x - x^2}.$

2. Simplify the highest degree terms in numerator and denominator: Divide both the numerator and denominator by $x^2$, the highest power of $x$ in the denominator: $\frac{\frac{4x^2}{x^2} + \frac{12x}{x^2} + \frac{9}{x^2}}{\frac{5}{x^2} - \frac{3x}{x^2} - \frac{x^2}{x^2}}.$

   This simplifies to: $\frac{4 + \frac{12}{x} + \frac{9}{x^2}}{\frac{5}{x^2} - \frac{3}{x} - 1}.$

3. Take the limit as $x \to \infty$: As $x \to \infty$, the terms $\frac{12}{x}$, $\frac{9}{x^2}$, and $\frac{5}{x^2}$ all approach 0. This leaves: $\frac{4}{-1} = -4.$

Final Answer:

$-4$

Let me know if you'd like further explanation or have more questions!

5 Related Questions:

1. How do you determine the highest degree term in a rational function?
2. What happens to lower degree terms when taking limits as $x \to \infty$?
3. How do you simplify rational functions with polynomials in the numerator and denominator?
4. What are some alternative methods for solving rational limits?
5. What is the importance of dividing by the highest degree term in these problems?

Tip:

Always focus on the terms with the highest degree when calculating limits at infinity; they dominate the behavior of the function.