We are tasked with finding the limits of the rational function $f(x) = \frac{x + 7}{x^3 + 11}$ as $x \to \infty$ and $x \to -\infty$.

Step 1: Analyze the degrees of the numerator and denominator

• The numerator, $x + 7$, has a degree of $1$.
• The denominator, $x^3 + 11$, has a degree of $3$.

Since the degree of the denominator is greater than the degree of the numerator, $f(x) \to 0$ as $x \to \pm\infty$.

Step 2: Confirm behavior at $x \to \infty$

For large positive $x$: $f(x) = \frac{x + 7}{x^3 + 11} \approx \frac{x}{x^3} = \frac{1}{x^2}.$ As $x \to \infty$, $\frac{1}{x^2} \to 0$.

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

Step 3: Confirm behavior at $x \to -\infty$

For large negative $x$: $f(x) = \frac{x + 7}{x^3 + 11} \approx \frac{x}{x^3} = \frac{1}{x^2}.$ Again, as $x \to -\infty$, $\frac{1}{x^2} \to 0$.

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

Final Answer:

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

Would you like a more detailed breakdown or explanation?

5 Related Questions:

1. How does the degree of the numerator and denominator affect the behavior of rational functions?
2. What happens if the degrees of the numerator and denominator are equal?
3. How do you determine limits at infinity for other types of functions (e.g., exponential or logarithmic)?
4. What if the denominator had a smaller degree than the numerator?
5. How do horizontal asymptotes relate to limits at infinity?

Tip:

When evaluating the limit of a rational function, always simplify by dividing numerator and denominator by the highest power of $x$ in the denominator.