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The image shows a limit problem. Here are the answers:

1. \(\lim_{{x \to \infty}} \frac{{x^2 + 9}}{{4x^2 + \sqrt{4x^4 + x^2 - 2}}}\):

   **Answer:** \( \boxed{0} \)

If you have further questions or need clarifications, feel free to ask!

### Further Practice Questions:
1. How can you prove the limit approaches 0 as \( x \to \infty \)?
2. What is the role of the square root term in the denominator for large \( x \)?
3. What happens if the highest powers of \( x \) are equal in both numerator and denominator?
4. How would the result change if the \( x^2 \) term in the numerator was multiplied by 2?
5. What limit rules simplify complex fractions like this?

**Tip:** For limits at infinity, comparing the highest degree terms in the numerator and denominator is often the best approach.

Evaluate \( \lim_{{x \to \infty}} \frac{{x^2 + 9}}{{4x^2 + \sqrt{4x^4 + x^2 - 2}}} \)

Solve the limit problem \( \lim_{{x \to \infty}} \frac{{x^2 + 9}}{{4x^2 + \sqrt{4x^4 + x^2 - 2}}} \) by analyzing the highest degree terms in the numerator and denominator. Learn about limits at infinity and rational function simplification.

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