The given problem is:

$\lim_{x \to \infty} \frac{x + 10}{x^2 + 11}$

To solve this limit, let's analyze it step-by-step:

Step 1: Simplify the expression

We will divide the numerator and the denominator by the highest power of $x$ present in the denominator, which is $x^2$.

$\frac{x + 10}{x^2 + 11} = \frac{\frac{x + 10}{x^2}}{\frac{x^2 + 11}{x^2}} = \frac{\frac{x}{x^2} + \frac{10}{x^2}}{1 + \frac{11}{x^2}}$

Simplify each term:

$= \frac{\frac{1}{x} + \frac{10}{x^2}}{1 + \frac{11}{x^2}}$

Step 2: Take the limit as $x \to \infty$

As $x \to \infty$, both $\frac{1}{x}$ and $\frac{10}{x^2}$ approach 0.

$\lim_{x \to \infty} \frac{\frac{1}{x} + \frac{10}{x^2}}{1 + \frac{11}{x^2}} = \frac{0 + 0}{1 + 0} = 0$

Final Answer:

$\lim_{x \to \infty} \frac{x + 10}{x^2 + 11} = 0$

Would you like any further details or have any questions?

Related Questions:

1. How do you find the limit of a rational function as $x \to -\infty$?
2. What happens if the degree of the numerator is greater than the degree of the denominator in limits?
3. How can L'Hopital's Rule be applied to solve this type of limit problem?
4. What is the importance of dividing by the highest power of $x$ in the denominator?
5. How do asymptotes relate to the behavior of rational functions as $x \to \infty$?

Tip:

When dealing with limits at infinity for rational functions, always compare the degrees of the numerator and the denominator to determine the limit quickly.