The problem asks to find the limit of the rational function

$f(x) = \frac{x + 7}{x^3 + 11}$

as $x \to -\infty$.

Solution:

1. Leading term analysis:

   • The numerator is $x + 7$. For large $x$, the leading term is $x$.
   • The denominator is $x^3 + 11$. For large $x$, the leading term is $x^3$.

   So, as $x \to -\infty$, the dominant behavior of $f(x)$ is determined by the ratio of the leading terms:

   $\frac{x}{x^3} = \frac{1}{x^2}.$

2. Behavior as $x \to -\infty$:

   • When $x \to -\infty$, $x^2$ grows large and positive, so $\frac{1}{x^2} \to 0^+$.

   Thus:

   $\lim_{x \to -\infty} f(x) = 0.$

Final Answer:

The limit of $f(x)$ as $x \to -\infty$ is 0.

Would you like a step-by-step breakdown or additional explanation?

Here are five related questions to consider:

1. What happens to $f(x)$ as $x \to \infty$ instead of $-\infty$?
2. How can the concept of horizontal asymptotes be applied to rational functions like this one?
3. What is the difference between limits at infinity and finite limits?
4. How do higher-degree terms in the numerator affect limits at infinity?
5. Can you determine if the function $f(x)$ is increasing or decreasing as $x \to -\infty$?

Tip:

Always compare the highest-degree terms in the numerator and denominator when evaluating limits at infinity for rational functions!